Self-induced avor conversion of supernova neutrinos on small … · 2016. 1. 18. · JCAP01(2016)028 ou rnal of C osmology and A strop ar ticle P hysics J An IOP and SISSA journal

JCAP01(2016)028

ournal of Cosmology and Astroparticle PhysicsAn IOP and SISSA journalJ

Self-induced flavor conversion ofsupernova neutrinos on small scales

S Chakrabortya R S Hansenb I Izaguirrea and GG Raffelta

aMax-Planck-Institut fur Physik (Werner-Heisenberg-Institut)Fohringer Ring 6 80805 Munchen GermanybDepartment of Physics and Astronomy University of Aarhus8000 Aarhus C Denmark

E-mail sovanmppmpgde rshansenphysaudk izaguirrmppmpgderaffeltmppmpgde

Received August 7 2015Accepted December 18 2015Published January 15 2016

Abstract Self-induced flavor conversion of supernova (SN) neutrinos is a generic featureof neutrino-neutrino dispersion The corresponding run-away modes in flavor space canspontaneously break the original symmetries of the neutrino flux and in particular can spon-taneously produce small-scale features as shown in recent schematic studies However theunavoidable ldquomulti-angle matter effectrdquo shifts these small-scale instabilities into regions ofmatter and neutrino density which are not encountered on the way out from a SN The tra-ditional modes which are uniform on the largest scales are most prone for instabilities andthus provide the most sensitive test for the appearance of self-induced flavor conversion Asa by-product we clarify the relation between the time evolution of an expanding neutrinogas and the radial evolution of a stationary SN neutrino flux Our results depend on severalsimplifying assumptions notably stationarity of the solution the absence of a ldquobackwardrdquoneutrino flux caused by residual scattering and global spherical symmetry of emission

Keywords core-collapse supernovas supernova neutrinos

ArXiv ePrint 150707569

Article funded by SCOAP3 Content from this work may be usedunder the terms of the Creative Commons Attribution 30 License

Any further distribution of this work must maintain attribution to the author(s)and the title of the work journal citation and DOI

doi1010881475-7516201601028

JCAP01(2016)028

Contents

1 Introduction 1

2 Equations of motion 421 Setting up the system 422 Large-distance approximation 523 Single angle vs multi angle 624 Two-flavor system 725 Mass ordering 926 Linearization 927 Homogeneous neutrino and electron densities 1028 Spatial Fourier transform 1129 Oscillation eigenmodes 11210 Monochromatic and isotropic neutrino distribution 11

3 One-dimensional system 1231 Single angle (v = plusmn1) 12

311 Eigenvalue equation 12312 Homogeneous mode (k = 0) 13313 Inhomogeneous modes (k gt 0) 13

32 Multi-angle effects (0 le v le 1) 14321 Eigenvalue equation 14322 Homogeneous mode (k = 0) without matter effects (λ = 0) 15323 Inhomogeneous modes (k gt 0) without matter effects (λ = 0) 18

33 Including matter(λ 6= 0) 19331 Homogeneous mode (k = 0) 19332 Inhomogeneous mode (k gt 0) 22

4 Two-dimensional system 2241 Single angle (|v| = 1) 22

42 Multi-angle effects (0 le v le 1) 26421 Eigenvalue equation 26422 Homogeneous mode (k = 0) with matter (λ gt 0) 26423 Inhomogeneous mode (k gt 0) without matter (λ = 0) 27424 Inhomogeneous mode (k gt 0) with matter (λ gt 0) 29

5 Conclusions 31

A Analytic integrals 32

B Frequently encountered eigenvalue equations 33

C Asymptotic solutions for 1D and λrarrinfin 36

ndash i ndash

JCAP01(2016)028

D Asymptotic solutions for 2D with λ = 0 and krarrinfin 39

E Asymptotic solutions for 2D with k = 0 and λrarrinfin 42

1 Introduction

Core-collapse supernovae (SNe) or neutron-star mergers are powerful neutrino sources andprobably the only astrophysical phenomena where these elusive particles are dynamicallyimportant and crucial for nucleosynthesis [1 2] The low energies of some tens of MeV implythat β reactions of the type νe +nharr p+ eminus and νe + pharr n+ e+ are the dominant charged-current processes Heavy-lepton neutrinos νmicro νmicro ντ and ντ in this context often collectivelyreferred to as νx interact only by neutral-current processes Therefore neutrino energytransfer and the emitted fluxes depend on flavor and one may think that flavor oscillationsare an important ingredient However the large matter effect in this dense environmentimplies that eigenstates of propagation and those of interaction very nearly coincide [3 4]In spite of large mixing angles flavor oscillations are irrelevant except for MSW conversionwhen neutrinos pass the resonant density as they stream away [5 6] Therefore the neutrinosignal from the next galactic SN may carry a detectable imprint of the yet unknown neutrinomass hierarchy [7ndash11]

This picture can fundamentally change when the refractive effect of neutrinos on eachother is included [12ndash14] The mean field representing background neutrinos can have fla-vor off-diagonal elements (ldquooff-diagonal refractive indexrdquo) due to flavor coherence caused byoscillations and can lead to strong flavor conversion effects [15ndash54] (We ignore additionaleffects that would arise from non-standard neutrino interactions [55] spin-flip effects causedby neutrino magnetic dipole moments [56ndash58] by refraction in inhomogeneous or anisotropicmedia [59ndash62] or the role of neutrino-antineutrino pair correlations [63ndash66]) Self-inducedflavor conversion preserves the global flavor content of the ensemble but re-shuffles it amongmomentum modes or between neutrinos and antineutrinos The simplest example would bea gas of νe and νe converting to νmicro and νmicro leaving the overall flavor content unchangedThe interacting modes of the neutrino field can be seen as a collection of coupled oscillatorsin flavor space The eigenmodes of this interacting system include collective harmonic os-cillations but can also include run-away solutions (instabilities) which lead to self-inducedflavor conversion [32] Under which physical conditions will instabilities occur how can wevisualize them and how will they affect the flavor composition of neutrinos propagating inthe early universe or stream away from a SN core

To study these questions many simplifications were used and especially symmetry as-sumptions were made to reduce the dimensionality of the problem However symmetryassumptions suppress those unstable solutions which break the assumed symmetry There-fore when instabilities are the defining feature of the dynamics symmetry assumptionsabout the solutions can lead to misleading conclusions because even if the system was setup in a symmetric state the interacting ensemble can break this symmetry spontaneouslyThis behavior is analogous to the hydrodynamical aspects of SN physics which cannot showconvective overturn if the simulation is spherically symmetric yet such 3D effects are nowunderstood to be crucial for SN physics [67ndash82]

Our present concern is the question of ldquospatial spontaneous symmetry breakingrdquo in self-induced flavor conversion In previous studies the flavor content of neutrinos streaming awayfrom a SN core was taken to remain uniform in the transverse directions However recent

ndash 1 ndash

JCAP01(2016)028

studies of simplified systems suggest that this symmetry can be spontaneously broken [51ndash54] To avoid the complication of global spherical coordinates it is sufficient to model theneutrino stream at some distance with plane-parallel geometry ie we can use wave vectorsk in the transverse plane to describe small-scale spatial variations In this terminologytraditional studies only considered k = 0 (global spherical symmetry) In analogy to thisk = 0 case it was found that for any k there is some range of effective neutrino densitieswhere unstable solutions exist [51ndash54] We usually express the neutrino density in terms ofan effective neutrino-neutrino interaction energy micro =

radic2GFnνe(r) (Rr)2 where nνe(r) is

the νe density at distance r and R is some reference radius playing the role of the neutrinosphere The parameter micro varies with rminus4 because the neutrino density decreases as rminus2 withdistance In this terminology for any k there is a range micromin lt micro lt micromax where the system isunstable For larger k (smaller spatial scales) the instability range shifts to larger micro ie toregions closer to the SN core This finding suggests that the neutrino stream is never stablebecause at any neutrino density there is some range of unstable k modes

However such conclusions may be premature as one also needs to include the refractiveeffect of matter which also tends to shift the instability to larger micro a phenomenon termedldquomulti-angle matter suppressionrdquo of the instability [26 37] We usually express the multi-angle matter effect in terms of the parameter λ =

radic2GFne(r) (Rr)2 where ne(r) is the

electron density at distance r One should study the instability region in the two-parameterspace of effective matter and neutrino densities λ and micro which we call the ldquofootprint of theinstabilityrdquo In figure 1 we show as an example the footprint of the MAA instability for aschematic SN model (MAA stands for ldquomulti azimuth anglerdquo ie one type of instability)We always define the instability region by the requirement that the growth rate κ gt 10minus2 ω0where ω0 is a typical vacuum oscillation frequency

Our schematic SN model consists of neutrinos and antineutrinos with a single energycorresponding to a vacuum oscillation frequency ω0 = ∆m22E = 1 kmminus1 We assume theyare emitted isotropically at the neutrino-sphere radius R = 30 km We consider two-flavoroscillations and after subtracting the νx flux we are left with a νe and νe flux such that thereemerge twice as many νe than νe Finally we assume that the effective neutrino-neutrinointeraction energy micro =

radic2GFnνe(Rr)

2 at the neutrino sphere r = R is micro0 = 105 kmminus1Under these circumstances run-away solutions for k = 0 exist within the footprint areashown in figure 1 as a blue-shaded region We also show as a solid red line a possible densityprofile corresponding to electron density as function of radius The sudden density drop atr asymp 200 km is the shock wave In this example the density profile does not intersect withthe instability footprint for radii below the shock wave On the other hand for larger radiicollective flavor conversion would begin We further show the footprint for inhomogeneitieswith assumed wave-number k = 102 and k = 103 measured in units of the vacuum oscillationfrequency ω0 Notice that we define k in ldquoco-movingrdquo coordinates along the radius ie afixed k represents a fixed angular scale not a fixed length scale Notice also that for theMAA instability shown here which is relevant for normal mass ordering non-zero k valueslead to two instability regions This phenomenon does not arise for the bimodal instability

The full range of all k-values inevitably fills the entire region below the k = 0 mode(blue shading) in this plot ie the entire gray-shaded region is unstable whereas the regionabove the blue-shaded part remains unscathed In other words essentially the largest-scalemode with k = 0 is the ldquomost dangerousrdquo mode If this mode is stable on the locus of the SNdensity profile in figure 1 the higher-k modes are stable as well Of course if any instabilityis encountered by the physical SN density profile these instabilities will span a range of scalesand create complicated flavor conversion patterns

ndash 2 ndash

JCAP01(2016)028

50 100 200 500 1000

10610-1100101102103104

Radius HkmL

Μ Hkm-1L

SNDensity

k = 0102103

Figure 1 Footprint of the MAA instability region in the parameter space of effective neutrino densitymicro =radic

2GFnνe(Rr)2 where R is the neutrino-sphere radius and matter density λ =radic

2GFne(Rr)2

for the schematic SN model described in the text Because micro prop rminus4 the horizontal axis is equivalentto the distance from the SN as indicated on the lower horizontal axis We also show a representativeschematic SN density profile where the sharp density drop marks the shock wave We also showthe instability footprint explicitly for co-moving wave numbers k = 102 and k = 103 in units of thevacuum oscillation frequency Notice that for the same value of k there are two separate instabilitystrips The collection of all small-scale instabilities fill the gray-shaded region below the traditionalk = 0 (blue shaded) instability region whereas they leave the space above untouched

The rest of our paper is devoted to substantiating this main point and to explain ourexact assumptions We stress that our simplifications may be too restrictive to provide areliable proxy for a realistic SN In particular we assume stationary neutrino emission andthat the solution is stationary as well ie we assume that the evolution can be expressedas a function of distance from the surface alone We also ignore the ldquohalo fluxrdquo caused byresidual scattering which can be a strong effect Our study would not be applicable at allin regions of strong scattering ie below the neutrino sphere We assume that the originalneutrino flux is homogeneous and isotropic in the transverse directions ie global sphericalsymmetry of emission at the neutrino sphere It has not yet been studied if this particularassumption has any strong impact on the stability question ie if violations of such anideal initial state substantially change the instability footprint or if such disturbances wouldsimply provide seeds for instabilities to grow It is impossible to understand and study alleffects at once so here we only attempt to get a grasp of the differential impact of includingspatial inhomogeneities in the form of self-induced small-scale flavor instabilities All theother questions must be left for future studies

ndash 3 ndash

JCAP01(2016)028

2 Equations of motion

Beginning from the full equation of motion for the neutrino density matrices in flavor spacewe develop step-by-step the simplified equations used in our linearized stability analysis Inparticular we formulate the stationary spherical SN problem where the flavor evolution isa function of radius as an equivalent time-dependent 2D problem in the tangential planeA fixed neutrino speed in the tangential plane corresponds to the traditional ldquosingle anglerdquotreatment whereas neutrino speeds taking on values between 0 and a maximum determinedby the distance from the neutrino sphere corresponds to the traditional ldquomulti anglerdquo case

21 Setting up the system

We describe the neutrino field in the usual way by 3times3 flavor matrices (t r Ev) where thediagonal elements are occupation numbers for the different flavors whereas the off-diagonalelements contain correlations among different flavor states of equal momentum We followthe convention where antineutrinos are described by negative energy E and the correspondingmatrix includes a minus sign ie it is a matrix of negative occupation numbers

We always work in the free-streaming limit ignoring neutrino collisions In this caseneutrino propagation is described by the commutator equation [14 45]

i(partt + v middotnablar) = [H ] (21)

where and H are functions of t r E and v The Hamiltonian matrix is

2E+radic

intdΓprime

(v minus vprime)2

2trEprimevprime

] (22)

where M2 the matrix of neutrino mass-squares is what causes vacuum oscillations Thematrix of charged-lepton densities N` provides the usual Wolfenstein matter effect Theintegration dΓprime is over the neutrino and antineutrino phase space Because antineutrinos aredenoted with negative energies we have explicitly

intdΓprime =

int +infinminusinfin dEprimeEprime2

intdvprime(2π)3 and the

velocity integration dvprime is over the unit sphere Because the neutrino speed |v| = 1 we wereable for later convenience to write the current-current velocity factor in the unusual form(1minus v middot vprime) = 1

2(v minus vprime)2

Studying this 7-dimensional problem requires significant simplifications For neutrinooscillations in the early universe one will usually assume initial conditions at some timet = 0 and then solve these equations as a function of time To include spatial variations onemay Fourier transform these equations in space replacing the spatial dependence on r by awave-number dependence k whereas v middotnablar rarr iv middot k and the rhs becomes a convolutionof Fourier modes [51] One can then perform a linearized stability analysis for every modek and identify when modes of different wave number are unstable and lead to self-inducedflavor conversion [52] One can also use this representation for numerical studies [51 53 54]

The other relatively simple case is inspired by neutrinos streaming from a supernova(SN) core One assumes that on the relevant time scales the source is stationary and that thesolution is stationary as well so that partt rarr 0 In addition one assumes that neutrinos streamonly away from the SN so that it makes sense to ask about the variation of the neutrinoflavor content as a function of distance assuming we are provided with boundary conditionsat some radius R which we may call the neutrino sphere Actually this description can bea poor proxy for a real SN because the small ldquobackwardrdquo flux caused by residual neutrino

ndash 4 ndash

JCAP01(2016)028

scattering in the outer SN layers the ldquohalo fluxrdquo can be surprisingly important for neutrino-neutrino refraction because of its broad angular range [40ndash42] Here we will ignore this issueand use the simple picture of neutrinos streaming only outward

We stress that this simplification is the main limitation of our study and its interpre-tation in the physical SN context If (self-induced) instabilities exist on small spatial scalesthey could even exist below the neutrino sphere where the picture of neutrinos streaming onlyin one direction would be very poor The approach taken here to reduce the 7-dimensionalproblem to a manageable scope may then hide the crucial physics Therefore our case studyleaves open important questions about a real SN

22 Large-distance approximation

The main point of our study is to drop the assumption of spatial uniformity ie we includevariations transverse to the radial direction However we are not interested in an exactdescription of large-scale modes At some distance from the SN outward-streaming neutrinoscannot communicate with others which travel in some completely different direction as longas we only include neutrino-neutrino refraction and not for example lateral communicationby hydrodynamical effects If we are only interested in relatively small transverse scaleswe may approximate a given spherical shell locally as a plane allowing us to use Cartesiancoordinates in the transverse direction rather than a global expansion in spherical harmonics

We now denote with z the ldquoradialrdquo direction and use bold-faced characters to de-note vectors in the transverse plane notably a for the coordinate vector in the transverseplane and β the transverse velocity vector In the stationary limit the equation of motion(EoM) becomes

i(vzpartz + β middotnablaa) = [H ] (23)

where vz =radic

1minus β2 and and H depend on z a E and β If the neutrino-sphere radiusis R then at distance r from the SN the maximum neutrino transverse velocity is βmax asympRr 1 This latter ldquolarge distance approximationrdquo is the very justification for usingCartesian coordinates in the transverse direction

Therefore it is self-consistent to expand the equations to order β2 (We need to go toquadratic order lest the neutrino-neutrino interaction term vanishes) The β expansion isnot necessary for our stability analysis but avoiding it does not provide additional precisionand performing it provides significant conceptual clarity Numerical precision for a specificSN model is not our goal and in this case we would have to avoid modeling the transversedirection as a flat space anyway especially when considering regions that are not very faraway from the SN core

In equation (23) we multiply with 1vz asymp (1 + 12β

2) and notice that the gradient termremains unchanged if we expand only to order β2 so that

i(partz + β middotnablaa) = [H ] (24)

where the Hamiltonian matrix is the old one times (1 + 12β

2) or explicitly

2E+radic

)+radic

intdΓprime

(β minus βprime)2

2zaEprimeβprime (25)

The flux factor under the integral in the second expression is also an expansion to O(β2) inthe form 1minus vzvprimezminusβ middotβprime = 1minus

radic1minus β2

radic1minus βprime2minusβ middotβprime asymp 1

2β2 + 1

2βprime2minusβ middotβprime = 1

2(βminusβprime)2Multiplying this expression with (1 + 1

2β2) makes no difference because it is already O(β2)

ndash 5 ndash

JCAP01(2016)028

As a next step we re-label our variables and denote the radial direction z as time tMoreover we rescale the transverse velocities as β = vβmax where v is now a 2D vectorobeying 0 le |v| le 1 Coordinate vectors in the transverse plane are also rescaled as x =aβmax ie the new transverse coordinate vector x is ldquoco-movingrdquo in that it denotes a fixedangular scale relative to the SN After these substitutions the EoMs are

i(partt + v middotnablax) = [H ] (26)

β2max

2E+radic

)+radic

2GFβ2max

intdΓprime

(v minus vprime)2

2txEprimevprime (27)

Of course the neutrino phase-space integrationintdΓprime is understood in the new variables

Our stationary 3D problem has now become a time-dependent 2D problem In the SNinterpretation the aspect ratio of the neutrino sphere shrinks with distance and correspond-ingly βmax shrinks In other words the physics is analogous to neutrino oscillations in theexpanding universe In the SN case linear transverse scales grow as rR where r is the dis-tance to the SN ie our ldquoHubble parameterrdquo is Rminus1 where R is the neutrino-sphere radiusand the scale factor grows linearly with ldquotimerdquo The physical neutrino density decreases withinverse-distance squared and in addition the factor β2

max accounts for the decreasing valueof the current-current factor in the neutrino interaction Therefore the effective neutrinonumber density decreases with (scale factor)minus4 in the familiar way

23 Single angle vs multi angle

In the SN context one often distinguishes between the single-angle and multi-angle casesreferring to the zenith angle of neutrino emission at the SN core If all neutrinos wereemitted with a fixed zenith angle their transverse speeds would be |β| = βmax and in ournew variables |v| = 1 In this case (1 + 1

2β2maxv2) rarr (1 + 1

2β2max) is simply a small and

negligible numerical correction to the vacuum oscillation frequencies and the matter effectAlso we can revert to the traditional form of the flux factor 1

2(v minus vprime)2 = (1 minus v middot vprime)Therefore the SN single-angle case is equivalent without restrictions to a 2D neutrino gasevolving in time Therefore neutrino oscillations in an expanding space (ldquoearly universerdquo) isexactly equivalent to the single-angle approximation of neutrinos streaming from a SN corewith properly scaled effective neutrino and matter densities

It has been recognized a long time ago that in the single-angle case the ordinary mattereffect has no strong impact on self-induced flavor conversion [19] As usual one can go to arotating coordinate system in flavor space In this new frame the matrix of vacuum oscillationfrequencies M22E has fast-oscillating off-diagonal elements and in a time-averaged senseit is diagonal in the weak-interaction basis These fast-oscillating terms are what kick-startsthe instabilities at the beginning of self-induced flavor conversion but are otherwise irrelevantFor a larger matter effect more e-foldings of exponential growth of the instability are neededto ldquogo nonlinearrdquo In this sense matter has a similar effect concerning the onset of theinstability that would be caused by reducing the mixing angle These effects concern theperturbations which cause the onset of instabilities not the existence and properties of theunstable modes themselves

We may ignore the small correction to the vacuum oscillation frequency provided bythe factor (1 + 1

2β2maxv2) We need to keep terms of order β2 in the context of the matter

ndash 6 ndash

JCAP01(2016)028

and neutrino-neutrino term which in the interesting case are large and after multiplicationwith β2

max still larger than the vacuum oscillation term Therefore we find

langM2

rang+radic

2GFN`β2max

2+radic

2GFβ2max

intdΓprime

(v minus vprime)2

where the first term symbolizes the time-averaged vacuum term in the fast co-rotating frameIn the single-angle case where v2 = 1 for all modes the remaining matter term can be rotatedaway as well

The multi-angle SN case in this representation corresponds to a 2D neutrino gas withvariable propagation speed 0 le |v| le 1 ie the velocity phase space is not just the surfaceof the 2D unit sphere (a circle with |v| = 1) but fills the entire 2D unit sphere (a disk with|v| le 1) There is no counterpart to this effect in a ldquonormalrdquo neutrino gas The early-universeanalogy does not produce multi-angle effects but we can include them without much ado byallowing the neutrino velocities to fill the 2D unit sphere

If neutrinos are emitted ldquoblack-body likerdquo from a spherical surface from a distance thisneutrino sphere looks like a disk of uniform surface brightness in analogy to the solar disk inthe sky Therefore this assumption corresponds to the neutrino transverse velocities fillingthe 2D unit sphere uniformly In earlier papers of our group we have used the variable u = v2

with 0 le u le 1 as a co-moving transverse velocity coordinate representing the neutrinozenith angle of emission In terms of this variable the black-body like case corresponds tothe familiar top-hat u distribution on the interval 0 le u le 1

Our overall set-up was inspired by that of Duan and Shalgar [52] except that theyconsider only one transverse dimension with single-angle emission at the SN In other wordstheir system is equivalent to two colliding beams evolving in time and allowing spatial vari-ations A numerical study of this case in both single and multi angle was very recentlyperformed by Mirizzi Mangano and Saviano [53] going beyond the linearized case Not un-expectedly the outcome of the nonlinear evolution is found to be flavor decoherence In ourapproach multi-angle effects in this colliding-beam system can be easily included by usinga 1D velocity distribution that fills the entire interval minus1 le v le +1 and not just the twovalues v = plusmn1 A black-body like zenith-angle distribution corresponds to a uniform velocitydistribution on this interval

24 Two-flavor system

As a further simplification we limit our discussion to a two-flavor system consisting of νe andsome combination of νmicro and ντ that we call νx following the usual convention in SN physicsWe are having in mind oscillations driven by the atmospheric neutrino mass difference andby the small mixing angle Θ13 The vacuum oscillation frequency is

ω =∆m2

2E= 063 kmminus1

(10 MeV

) (29)

where we have used ∆m2 = 25times 10minus3 eV2 Henceforth we will describe the neutrino energyspectrum by an ω spectrum instead with negative ω describing antineutrinos

The matrix of vacuum oscillation frequencies in the fast-rotating flavor basis takes onthe diagonal form lang

rangrarr ω

2 00 minus1

) (210)

ndash 7 ndash

JCAP01(2016)028

where we have removed the part proportional to the unit matrix which drops out of com-mutator expressions We do not include the fast-oscillating off-diagonal elements which isirrelevant for the stability analysis The matter effect appears in a similar form

radic2GFN`β

2rarr λv2

2 00 minus1

) (211)

where again we have removed the piece proportional to the unit matrix The parameterλ describing the multi-angle matter effect at distance r from the SN with neutrino-sphereradius R and using βmax = Rr is

λ =radic

2GFne(r)R2

r2= 386times 108 kmminus1 Ye(r) ρ(r)

1012 g cmminus3

r2 (212)

where ne(r) is the net density of electrons minus positrons ρ(r) the mass density and Ye(r)the electron fraction per baryon each at radius r The matter density drops steeply outsidethe neutrino sphere and jumps downward by an order of magnitude at the shock-wave radiusTherefore we need to consider λ values perhaps as large as some 107 kmminus1 all the way tovanishingly small values

Turning to the neutrino-neutrino term notice that the matrices play the role ofoccupation numbers and that the

intdΓ integration includes the entire phase space of occupied

neutrino and antineutrino modes Therefore Nν =intdΓ is a flavor matrix of net neutrino

minus antineutrino number densities in analogy to the corresponding charged-lepton matrixN` It is less obvious however how to best define an effective neutrino-neutrino interactionstrength micro which plays an analogous role to λ If we were to study a system that initiallyconsists of equal number densities of νe and νe the matrix Nν vanishes but later developsoff-diagonal elements Therefore we rather use the number density of νe without subtractingthe antineutrinos and define

micro =radic

2GFnνe(r)R2

r2= 472times 105 kmminus1 Lνe

4times1052 ergs

10 MeV

〈Eνe〉

(30 km

)2 (Rr

where Lνe is the νe luminosity and 〈Eνe〉 their average energy More precisely nνe is the νedensity at radius r that we would obtain in the absence of flavor conversions after emissionat radius R Previously we have sometimes normalized micro to nνe instead or to the differencebetween the νe and νx densities However in our schematic studies we assume that initiallywe have only a gas consisting of νe and νe again obviating the need for these fine distinctionsThe exact definition of micro has no physical impact because it always appears as a product withthe density matrices

In previous papers [32 45] a further factor 12 was included in the definition of themulti-angle λ and micro We have kept this factor explicitly in equation (28) both in the matterterm and in the flux factor 1

2(v minus vprime)2 to maintain its traditional form In this way theequations can be directly applied to a traditional ldquoearly universerdquo system To make contactwith previous SN discussions one can always absorb this factor in the definition of λ and micro

As a next step we project out the trace-free part of the density matrices and normalizethem to account for the above normalization of the effective neutrino-neutrino interactionstrength micro

txωv =Tr(txωv)

2+nνe2

Gtxωv (214)

ndash 8 ndash

JCAP01(2016)028

With these definitions the two-flavor EoMs finally become

i(partt + v middotnablax)Gtxωv = [HtxωvGtxωv] (215)

with the Hamiltonian matrix

Htxωv =(ω + λx

12v2)(+1

2 00 minus1

)+ micro

intdΓprime

(v minus vprime)2

Gtxωprimevprime

2 (216)

where we have included a possible spatial dependence of the electron density in the form ofλx depending on location in the 2D space The neutrino velocity domain of integration isdetermined by the dimensionality of the chosen problem and if multi-angle effects are to beconsidered

25 Mass ordering

In a two-flavor system one important parameter for matter effects in general and for self-induced flavor conversion in particular is the mass ordering In our context the question isif the dominant mass component of νe is the heavier one (inverted ordering) or the lighterone (normal ordering) Traditionally ldquomass orderingrdquo is also termed ldquomass hierarchyrdquo andwe denote the two cases as IH (inverted hierarchy) and NH (normal hierarchy) We areconcerned with 1-3-mixing the corresponding mixing angle is not large and so it is clearwhat we mean with the ldquodominant mass componentrdquo

Our equations are formulated such that they apply to IH the traditional case whereself-induced flavor conversion is important in the form of the bimodal instability Of courseit has become clear that NH is actually the more interesting case For NH ∆m2 is negativebut we prefer to consider ∆m2 a positive parameter Therefore NH is achieved by includingexplicitly a minus sign on the rhs of equation (210) This change of sign translates into aminus sign for ω in the first bracket in equation (216)

For flavor conversion it is irrelevant if neutrinos oscillated ldquoclockwiserdquo or ldquocounterclockwiserdquo in flavor space ie in equation (215) we may change irarr minusi or Hrarr minusH withoutchanging physical results However the relative sign between ω and λ and micro is crucialTherefore switching the hierarchy is achieved by

IHrarr NH λrarr minusλ and microrarr minusmicro (217)

In our stability analysis we will consider the parameter range minusinfin lt micro lt +infin and minusinfin ltλ lt +infin as these are simply formal mathematical parameters Physically both parametersbeing positive corresponds to IH whereas the quadrant of both parameters being negativecorresponds to NH

26 Linearization

As a next step we linearize the EoMs in the sense that the complex off-diagonal element ofevery G is supposed to be very small compared to its diagonal part We write these matricesexplicitly as

(g GGlowast minusg

)(218)

ndash 9 ndash

JCAP01(2016)028

where g is a real and G a complex number and all quantities carry indices (tx ωv) Tolinear order in G we then find the EoMs

i(partt + v middotnablax) gtxωv = 0 (219a)

i(partt + v middotnablax)Gtxωv =

[ω + λx

12v2 + micro

intdΓprime 1

2(v minus vprime)2 gtxωprimevprime

]Gtxωv

minus gtxωv[micro

intdΓprime 1

2(v minus vprime)2Gtxωprimevprime

] (219b)

Up to normalization the ldquospectrumrdquo gtxωv is essentially the phase-space density of allneutrinos It is not affected by flavor conversion but evolves by free-streaming if it is nothomogeneous

27 Homogeneous neutrino and electron densities

We are primarily interested in self-induced instabilities Disturbances in the neutrino densityandor the electron density will certainly exist in a real SN and can play the role of seedsfor growing modes However if these disturbances are small it is unlikely that they willbe responsible for instabilities themselves Henceforth we will assume that the neutrino andelectron densities do not depend on the transverse coordinate x although the flavor contentmay well depend on x Free streaming does not change the density if it is uniform As aconsequence gtxωv does not depend on x or t and likewise λx does not depend on x

With this assumption gωv describes the initially prepared neutrino distribution ietheir density in the phase space spanned by ω and v Inspecting the first integral in equa-tion (219b) we may write the three independent terms as

intdΓ gωv ε1 =

intdΓ gωv v and ε2 =

intdΓ gωv v2 (220)

Here ε represents the ldquoasymmetryrdquo between neutrinos and antineutrinos The second termε1 represents a neutrino current which exists if their distribution is not isotropic and notsymmetric between neutrinos and antineutrinos Overall the first term in square bracketsbecomes

ω + 12λv2 + 1

2εmicrov2 minus microε1 middot v + 12ε2micro (221)

The last term is simply a constant and can be removed by changing the overall frequency ofthe rotating frame Defining

λ = λ+ εmicro (222)

the term in square brackets effectively becomes ω + 12 λv2 minus micro ε1 middot v We may also return to

the notation used in our previous papers and define

Stxωv =Gtxωvgωv

The linearized EoM then takes on the more familiar form

i(partt + v middotnablax)Stxωv =(ω + 1

2 λv2 minus micro ε1 middot v)Stxωv minus micro

intdΓprime 1

2(v minus vprime)2 gωprimevprimeStxωprimevprime

ndash 10 ndash

JCAP01(2016)028

This equation corresponds to equation (6) of reference [45] Besides the streaming term(the gradient term on the lhs) that we have now included to deal with self-induced in-homogeneities we have also found the additional term microε1 middot v which is unavoidable in anon-isotropic system irrespective of the question of homogeneity This neutrino flux term ismissing in reference [45] The presence of this term modifies the eigenvalue equation for anon-isotropic system

28 Spatial Fourier transform

We can now perform the spatial Fourier transform of our linearized EoM of equation (224)It simply amounts to replacing the spatial dependence on x of S by it dependence on thewave vector k and v middotnablax rarr iv middot k leading to

iStkωv =(ω + 1

2 λv2 + k middot v)Stkωv minus micro

intdωprime

intdvprime 1

2(v minus vprime)2 gωprimevprimeStkωprimevprime (225)

where k = kminus microε1 Therefore the wave vector k has the same effect as a neutrino currentIncluding an electron current would appear in a similar way However in the following studiesof explicit cases we will focus on isotropic distributions and worry primarily about self-inducedanisotropies not about the modifications caused by initially prepared anisotropies

In equation (225) we have spelled out the meaning of the phase-space integralintdΓprime =int

dωprimeintdv Notice that the meaning of

intdΓprime has changed in the course of changing variables

that describe the neutrino modes All phase-space factors and Jacobians have been absorbedin the definition of the effective neutrino-neutrino interaction strength micro as well as the nor-malization of the ldquospectrumrdquo gωv In particular if we begin with an ensemble consisting ofonly νe and νe and no νx or νx then our normalizations mean that

intinfin0 dω

intdv gωv = 1 In

this latter integral we have only included positive frequencies (neutrinos no antineutrinos)so that this normalization coincides with our definition that micro is normalized to nνe

29 Oscillation eigenmodes

In order to find unstable modes we seek solutions of our linearized EoM of the form Stkωv =QΩkωve

minusiΩt leading to an EoM in frequency space of the form

(12 λv2 + k middot v + ω minus Ω

)QΩkωv = micro

intdωprime

intdvprime 1

2(v minus vprime)2 gωprimevprime QΩkωprimevprime (226)

Eigenvalues Ω = γ + iκ with a positive imaginary part represent unstable modes with thegrowth rate κ

210 Monochromatic and isotropic neutrino distribution

In our explicit examples we will always consider monochromatic neutrinos with some fixedenergy implying a spectrum of two oscillation frequencies ω = plusmnω0 Assuming that theenergy and velocity distribution factorize we may write the spectrum in the form

gωv = hω fv (227)

The monochromatic energy spectrum is

hω = minusα δ(ω + ω0) + δ(ω minus ω0) (228)

ndash 11 ndash

JCAP01(2016)028

meaning that we have α antineutrinos (frequency ω = minusω0) for every neutrino (ω = ω0)The spectral asymmetry is ε = 1minus α

We will consider isotropic velocity distributions which in addition are uniform cor-responding to blackbody-like angular emission in the SN context In this case fv = 1Γvwhere Γv is the volume of the velocity phase space The eigenvalue equation (226) finallysimplifies to the form in which we will use it(

12 λv2 + k middot v + ω minus Ω

)QΩkωv = micro

intdωprime hωprime

intdvprime (v minus vprime)2QΩkωprimevprime (229)

We now consider systematically different cases of velocity distributions

3 One-dimensional system

As a first case study we consider a 1D system ie the toy model of ldquocolliding beamsrdquo thathas been used in the recent literature as a simple case where one can easily see the impact ofspontaneous spatial symmetry breaking [46 51ndash53] We go beyond previous studies in thatwe include the multi-angle matter effect and study the ldquofootprintrdquo of the various instabilitiesin the two-dimensional parameter space minusinfin lt micro lt +infin and minusinfin lt λ lt +infin This schematicstudy already leads to the conclusion that essentially the largest-scale instabilities are ldquomostdangerousrdquo in the context of SN neutrino flavor conversion

31 Single angle (v = plusmn1)

311 Eigenvalue equation

We begin with 1D systems ie colliding beams of neutrinos and antineutrinos with differentvelocity distributions The first case is what we call ldquosingle anglerdquo a nomenclature whichrefers to the zenith-angle distribution of SN neutrinos As we have explained in our wayof writing the equations ldquosingle anglerdquo means that the neutrino velocity distribution has|v| = 1 In our first 1D case this means we consider two colliding beams with v = plusmn1Matter effects can be rotated away

The eigenfunction QΩkωv now consists of four discrete components We denote thesefour amplitudes with the complex numbers R for right-moving (v = +1) neutrinos (ω =+ω0) R for right-moving antineutrinos and analogous L and L for left movers Our masterequation (229) then reads

ω0 + k 0 minusmicro microα

0 minusω0 + k minusmicro microαminusmicro microα ω0 minus k 0minusmicro microα 0 minusω0 minus k

minus Ω

= 0 (31)

corresponding to the equivalent result of Duan and Shalgar [52] The eigenvalues Ω are foundfrom equating the determinant of the matrix in square brackets with zero This conditioncan be written in the form(

minusk + ω0 minus Ωminus α

minusk minus ω0 minus Ω

k + ω0 minus Ωminus α

k minus ω0 minus Ω

)micro2 = 1 (32)

This expression depends only on micro2 and therefore yields identical eigenvalues for positive andnegative micro ie for both neutrino mass hierarchies as noted by Duan and Shalgar It is alsoeven under k rarr minusk as it must because the system was set up isotropically so the eigenvaluescannot depend on the orientation of k

ndash 12 ndash

JCAP01(2016)028

312 Homogeneous mode (k = 0)

For the homogeneous mode k = 0 this eigenvalue equation simplifies considerably Wealready know that we have the same solution for positive and negative micro where the latter isthe left-right symmetry breaking solution discovered in reference [46] We here limit ourselvesto micro gt 0 and need to solve the quadratic equation

ω20 minus Ω2 minus micro

[(1 + α)ω0 + (1minus α)Ω

]= 0 (33)

It has the solutions

Ω = minus(1minus α)micro

2plusmn

radic[ω0 +

(1minus α)micro

minus 2microω0 (34)

Unstable solutions exist for

(1 +radicα)2

ltmicro

(1minusradicα)2

which for α = 12 is the range 12minus8radic

2 lt microω0 lt 12 + 8radic

2 or numerically 06863 microω0 2331 The maximum growth rate is

κmax =2radicα

1minus αω0 (36)

For α = 12 this is κmaxω0 = 2radic

2 asymp 2828 The maximum growth rate occurs at theinteraction strength

microκmax =2 (1 + α)

(1minus α)2 (37)

We show the growth rate normalized to its maximum in figure 2 as a function of micromicroκmax For this normalization the unstable range is

1minus 2radicα

1 + αlt

microκmax

lt 1 +2radicα

1 + α (38)

If we use α = 1minus ε and expand to lowest order in ε this range is ε28 lt micromicroκmax lt 2minus ε28Therefore even if ε is not very small (ε = 12 in figure 2) the unstable range is close to itsmaximum range from 0 to 2

313 Inhomogeneous modes (k gt 0)

The quartic eigenvalue equation (32) is not easy to disentangle However for large k it sim-plifies and can be solved We may guess that for large k the real part of Ω is approximatelyminusk and without loss of generality we may go to a rotating frame such that Ω = Ω minus kMoreover based on numerical studies Duan and Shalgar [52] have conjectured that for largek the unstable micro-range scales with

radick This observation motivates us to write micro = m

radicω0k

and without loss of generality the eigenvalue equation reads(ω0

ω0 minus Ωminus αω0

minusω0 minus Ω

2k + ω0 minus Ωminus αk

2k minus ω0 minus Ω

)m2 = 1 (39)

ndash 13 ndash

JCAP01(2016)028

Figure 2 Growth rate κ for the unstable mode in the homogeneous (k = 0) 1D case for α = 12The maximum growth rate κmax is given in equation (36) the corresponding interaction strengthmicroκmax in equation (311)

Now we can take the limit k rarr infin and approximate the second bracket as (1 minus α)2 Theresulting quadratic equation is now easily solved and yields

ω0= minusε

4plusmnradic

16minus 8m2(2minus ε)ε+ ε4m4

4 (310)

One can try the same exercise with the opposite rotating frame Ω = Ω + k and finds asimilar-looking result where however the argument of the square-root is always positive iethere is no unstable mode with this property

From equation (32) one finds that the maximum growth rate κmax is the same as inthe homogeneous case of equation (36) ie for large k the maximum growth rate does notdepend on k and is the same as for k = 0 However it now occurs at the interaction strength

micro2κmax

=4 (1 + α)

(1minus α)3ω0k (311)

The unstable range is the same as given in equation (38) if we substitute micromicroκmax with(micromicroκmax)2 The growth rate as a function of micro is the same as shown in figure 2 if weinterpret the horizontal axis as (micromicroκmax)2 with our new microκmax

For intermediate values of k the maximum growth rate deviates slightly from the twoextreme cases It is somewhat surprising that the structure of the eigenvalue equation is suchthat the maximum possible growth rate depends only on the vacuum oscillation frequencyω0 and α but not on the potentially large scale k

32 Multi-angle effects (0 le v le 1)

We may now study the multi-angle impact of matter in the spirit of the SN system in our1D model by extending the velocity integration over the entire interval minus1 le v le 1 insteadof using only the modes v = plusmn1 One approach is to introduce nv discrete velocities onthe interval 0 lt v le 1 ie 2nv modes on the interval minus1 le v le 1 leading to a total of

ndash 14 ndash

JCAP01(2016)028

4nv discrete equations One may then proceed to find numerically the eigenmodes Thisapproach is marred by the appearance of spurious instabilities and one may need a largenumber of modes to obtain physical results [43]

Therefore we usually avoid discrete velocities and represent the eigenfunctions QΩkωv

as continuous functions of their variables Equation (229) reads for our specific case(12 λ v

2 + k v + ω minus Ω)QΩkωv = micro

int +infin

minusinfindωprime hωprime

int +1

minus1dvprime (v minus vprime)2QΩkωprimevprime (312)

Since our system was prepared ldquoisotropicrdquo in the sense of left-right symmetry the instabilitiescannot depend on the sign of k so that it is enough to consider 0 le k lt infin The rhs ofequation (312) as a function of v has the form A0 +A1v +A2v

2 Because 1 v and v2 arelinearly independent functions on the interval minus1 le v le +1 the lhs must be of that formas well and we may use the ansatz

QΩkωv =A0 +A1v +A2v

12 λ v

2 + k v + ω minus Ω(313)

for the eigenfunctions Inserting this form on both sides yields

A0 +A1v +A2v2 =

int +infin

int +1

minus1dvprime (vprime2 minus 2vvprime + v2)

A0 +A1vprime +A2v

prime2

12 λ v

prime2 + k vprime + ωprime minus Ω (314)

This equation really consists of three linearly independent equations for the parts proportionalto different powers of v so we get three equations linear in A0 A1 and A2 This set of linearequations is compactly written as1minus

I2 I3 I4

minus2I1 minus2I2 minus2I3

I0 I1 I2

= 0 (315)

In =micro

int +infin

minusinfindω hω

int +1

minus1dv

12 λ v

2 + k v + ω minus Ω (316)

where we have dropped the prime from the integration variablesNotice that In is odd under k rarr minusk if n is odd and it is even if n is even Moreover in

the determinant of the matrix in square brackets in equation (315) every term involving Inwith an odd power of n involves another factor Im with m odd Therefore the determinant iseven under k rarr minusk in agreement with our earlier statement that without loss of generalitywe may assume k ge 0

322 Homogeneous mode (k = 0) without matter effects (λ = 0)

As a first simple example we consider homogeneous solutions (k = 0) in the absence ofmatter effects (λ = 0) The latter assumption requires an exact cancellation λ = λ+ εmicro = 0between the matter effects caused by the background medium and by neutrinos themselvesWe consider this case only for mathematical convenience without physical motivation Thevelocity integrals vanish for odd powers of v For even powers and using the monochromaticfrequency spectrum of equation (228) we find

In =micro

2(1 + n)

ω0 + Ω+

ω0 minus Ω

2(1 + n)

(1 + α)ω0 minus (1minus α) Ω

ω20 minus Ω2

ndash 15 ndash

JCAP01(2016)028

and the eigenvalue equation corresponds to

13 0 1

50 minus2

3 01 0 1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (318)

Before searching for solutions we may diagonalize the 3times3 matrix This leads to threeindependent equations of the form of equation (33) where we need to substitute

microrarr micro

30times

radic5 gt 0

minus10 lt 0

5minus 3radic

5 lt 0

Therefore we get three instabilities one for micro gt 0 the usual bimodal instability (IH) andtwo negative-micro solutions (NH) The maximum growth rate is the same in every case as the onethat was found in section 312 and was given in equation (36) The exact unstable micro-rangeshave changed according to the micro scaling provided by equation (319) For our usual examplewith α = 12 the instability ranges for both hierarchies in the colliding-beam example were068 lt |microω0| lt 2331 After v-integration they become explicitly

176 lt microω0 lt 5974 (320a)

minus6994 lt microω0 lt minus206 (320b)

minus40944 lt microω0 lt minus1205 (320c)

We conclude that integrating over the velocity interval minus1 le v le +1 modifies the unstable micro-ranges breaks the symmetry between normal and inverted hierarchy and introduces anothernormal-hierarchy instability

Qualitatively these results are analogous to the three types of instability discoveredin the study of axial-symmetry breaking in the SN context [45] The one inverted-hierarchysolution appearing in all cases is the bimodal instability and corresponds to the original flavorpendulum [21] The first normal-hierarchy instability is what was termed the multi-azimuthangle (MAA) instability although in our 1D system we have only two ldquoazimuth anglesrdquo iethe two beam directions The third instability appearing in normal hierarchy for a muchlarger micro-range is what was termed the ldquomulti zenith anglerdquo (MZA) instability It requires inthe SN terminology a nontrivial range of zenith angles corresponding here to a non-trivialv-range ie anything beyond the trivial v = plusmn1 velocity distribution

Notice that our 1D MAA instability corresponds to an eigenfunction which is anti-symmetric in v (it breaks the left-right symmetry) and corresponds to the middle entry minus23in the matrix of equation (318) which decouples from the remaining 2times2 block The latteryields left-right symmetric solutions (even under v rarr minusv) In this sense it is the MZAinstability which corresponds for normal hierarchy to the bimodal solution It exists onlyfor at least three velocity modes and of course requires the presence of two vacuum oscillationfrequencies here always chosen as ω = plusmnω0 ie we need at least a total of six modes makinga simple visual interpretation more difficult

The growth rates of all three intabilities as functions of micro are shown in figure 3 as bluegreen and orange lines all of them having the same maximum In the top panel we overlaythe two instability curves for the original colliding-beam example where v = plusmn1 Thereforethis panel directly illustrates the effect of the velocity integration (ldquomulti-angle effectsrdquo in

ndash 16 ndash

JCAP01(2016)028

GrowthRateκ

k = 0 λ = 0 nv = 1

GrowthRateκ

k = 0 λ = 0 nv = 2

-500 -400 -300 -200 -100 0 1000

Interaction Strength μ

GrowthRateκ

k = 0 λ = 0 nv = 10

Figure 3 Growth rate κ for the unstable modes in the homogeneous (k = 0) 1D case with α = 12where micro lt 0 corresponds to normal and micro gt 0 to inverted hierarchy (Both κ and micro are in unitsof ω0) The colored curves common to all panels are the three instabilities which obtain after velocityintegration (minus1 le v le +1) with the instability ranges of equation (320) The overlaid black instabilitycurves are for discrete velocity bins where the number of bins nv = 1 2 and 10 as indicated in thepanels The overlay in the top panel (nv = 1) is the colliding-beam example with v = plusmn1 Withincreasing nv (top to bottom) discrete velocity bins approximate a uniform distribution

SN terminology) in that the velocity integration takes us from the two black curves to thethree colored ones

We have also studied the equations using a set of discrete velocities where the v = plusmn1case is the simplest example with nv = 1 bins (We count the number of bins in the range0 lt v le 1 ie there are equally many bins for negative velocities and the total numberdoubles for our two frequencies ω = plusmnω0) Adding the intermediate values v = plusmn12 takesus to nv = 2 shown in the second panel of figure 3 It reveals that the symmetry betweenthe hierarchies (micro rarr minusmicro symmetry) is broken as soon as the velocity range is non-trivialand that there are two normal-hierarchy solutions Increasing nv eventually approximatesa uniform v distribution A fairly small number of velocity bins is enough to achieve goodagreement We will see shortly that including non-zero k andor λ changes the picturebecause spurious instabilities appear

ndash 17 ndash

JCAP01(2016)028

-800 -600 -400 -200 00

GrowthRateκ

5101520

-150 -100 -50 0 50 100 1500

GrowthRateκ

1010 2020

-3000 -2500 -2000 -1500 -1000 -500 00

GrowthRateκ

255075100

-400 -200 0 200 4000

GrowthRateκ

2525 5050 7575 100100

Figure 4 Growth rates κ for different wave numbers k as indicated above the curves The otherparameters are λ = 0 and α = 12 The left panels show only the MZA mode in the right panels weshow the MAA mode (micro lt 0) and the bimodal mode (micro gt 0) The effect of non-zero k is essentiallyto shift the curves and for large k the instability curves are similar as a function of microk

323 Inhomogeneous modes (k gt 0) without matter effects (λ = 0)

Non-vanishing matter effects (λ 6= 0) and non-vanishing inhomogeneities (k 6= 0) modify theeigenvalue equation in similar ways the range of effective oscillation frequencies given by12 λv

2 + kv + ω increases considerably if kω0 1 andor λω0 1 Roughly one wouldsuspect that significant collective phenomena require a neutrino-neutrino coupling exceedingthis range of frequencies ie a micro range exceeding something like the rms spread of thisrange In this sense one would expect that the micro-range of unstable solutions would beshifted roughly linearly with λ andor k

We first test this picture with λ = 0 and k gt 0 The v-integrals in equation (316) canbe performed analytically (appendix A) and the ω-integration amounts to summing over twoterms with ω = plusmnω0 However finding the eigenvalues of equation (315) requires numericaltools We use Mathematica and show the result in figure 4 for 0 le kω0 le 100 as indicatedabove the curves In the left panels we only show the MZA mode For relatively small k thefunction shifts left and deforms somewhat whereas for larger k values it shifts left nearlylinearly with k We have checked that this linear behavior obtains numerically even for verylarge k values mdash we have tested values up to 107 In other words for large k the instabilitycurves are very similar as a function of microk although they narrow somewhat In contrast tothe earlier two-beam example there does not seem to be quite a universal function for largek Moreover in the earlier case the scaling was with micro

radicω0k ie the nontrivial v range

has qualitatively changed the results with regard to the k-scaling

ndash 18 ndash

JCAP01(2016)028

In the right panels of figure 4 we show analogous results for the MAA mode (micro lt 0)and the bimodal mode (micro gt 0) which show analogous behavior

One can study the same case by solving the eigenvalue equation in terms of velocitybins in analogy to what one would do if one were to solve the EoMs numerically insteadof performing only a linear stability analysis For k = 25ω0 we show the growth rates forall solutions in figure 5 for different choices for the number nv of velocity bins (We recallthat in our convention the total number of positive and negative v bins is 2nv) The largenumber of spurious modes is a conspicuous feature of these plots although for sufficientlylarge nv the physical modes stick out

With non-vanishing k andor λ the functional form of the eigenfunctions in equa-tion (313) no longer factorizes as a function of v and one of ω It is conceivable thatspurious modes can be avoided or their impact reduced if one were to find a better way ofdiscretizing the neutrino modes than by simple bins in velocity and frequency [44]

It is noteworthy that for all modes spurious or physical the growth rates are of orderω0 ie they do not inherit a larger frequency scale from k or in later cases from λ Evenfor huge values of k and λ this conclusion does not change and agrees with our explicit resultin the two-beam example

33 Including matter (λ 6= 0)

Including matter in our ldquomulti-zenith-anglerdquo case has the effect of introducing both λ andk in the denominator of the integrals of equation (316) They can still be done analytically(appendix A) but lead to transcendental functions Of course numerically one can find theeigenvalues without much problem The parameter λ like k has the effect of broadeningthe effective range of oscillation frequencies and of shifting the unstable collective modes tolarger values of |micro| We study the homogeneous and inhomogeneous cases separately

For the homogeneous mode (k = 0) the eigenvalue equation (315) simplifies considerablybecause in this case I1 = I3 = 0 and we are left with1minus

I2 0 I4

0 minus2I2 0I0 0 I2

= 0 (321)

We need to solve the two equations

(I2 minus 1)2 = I0I4 and I2 = minus12 (322)

As an overview we show in figure 6 a contour plot of the growth rate κ in the two-dimensionalparameter space of the interaction strength micro and the effective matter density λ = λ+ εmicro

As explained earlier the first quadrant (micro gt 0 and λ gt 0) corresponds physically toinverted mass ordering (IH) whereas the third quadrant (micro lt 0 and λ lt 0) correspondsto normal mass ordering (NH) The other quadrants would be relevant for example for abackground medium of antimatter Mathematically micro and λ are simply parameters whichwe leave unconstrained by physical considerations As usual we use α = 12 and thereforeε = 1minus α = 12 so that matter-free space (λ = 0) corresponds to the line λ = micro2

For λ = 0 the growth rates as a function of micro were shown in figure 3 in the formof the colored curves The effect of increasing |λ| is to shift the unstable regions to larger

ndash 19 ndash

JCAP01(2016)028

GrowthRateκ

k = 25 nv = 2

GrowthRateκ

k = 25 nv = 5

GrowthRateκ

k = 25 nv = 10

-1200-1000 -800 -600 -400 -200 0 2000

GrowthRateκ

k = 25 nv = 20

Figure 5 Growth rates for all modes that appear when we consider nv positive and nv negative-vbins using k = 25ω0 λ = 0 and α = 12 For growing nv more and more spurious modes appearbut their growth rates become smaller and the physical modes begin to stick out

ndash 20 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

EffectiveMatterDensity

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

Figure 6 Growth rate κ of the 1D instabilities as a function of micro and λ assuming α = 12 Upperpanel the first block in the eigenvalue equation (322) yields the bimodal instability for micro gt 0 andthe multi-zenith-angle (MZA) instability for micro lt 0 Lower panel the second block in equation (322)provides the multi-azimuth-angle (MAA) instability for micro lt 0 Notice that the first quadrant (micro λ gt0) represents IH the third quadrant (micro λ lt 0) represents NH

values of |micro| creating the butterfly image seen in figure 6 For micro gt 0 we obtain the bimodalinstability which exists even in a single-angle treatment For micro lt 0 we have two multi-angleinstabilities as indicated The solutions shown in the upper panel derive from the first blockin the eigenvalue equation (322) providing the bimodal and MZA instabilities The MAAsolution shown in the lower panel derives from the second block in equation (322)

The contours for the different instabilities are quite different In the regime of largeλ one can expand the eigenvalue equations in powers of 1λ and identify analytically thebehaviour of the footprint of the instability regions in the micro-λ-plane (appendix C) We showthe footprint of the contours of figure 6 on a logarithmic scale in appendix C in figure 15together with the asymptotic large-λ expansion

ndash 21 ndash

JCAP01(2016)028

332 Inhomogeneous mode (k gt 0)

We next determine numerically the same instability footprints for non-vanishing wave num-bers k We already know the impact of nonzero k for λ = 0 ie the instability is shiftedto larger micro values We do not expect a big difference for λ k relative to the k = 0 caseThese expectations are borne out by our results shown in figure 7 Considering first thesimpler micro gt 0 half of the plot we show the instability footprints for k = 0 102 103 and 104

as indicated in the plot We can now easily diagnose the impact of small-scale instabilitiesessentially they fill in the entire space between the k = 0 footprints between the quadrantwith positive and negative λ whereas the region above the k = 0 footprint in the upperquadrant and the space below in the lower quadrant remains stable The only small caveatis that in the upper quadrant for k sim λ there are ldquonosesrdquo of the footprint sticking into thepreviously stable region So there is a narrow sliver of parameters above the k = 0 footprintthe envelope of the noses which becomes unstable due to small-scale instabilities

In the left half of the plot (micro lt 0) the situation is somewhat more complicated becauseof the presence of two instabilities For large k the footprint actually connects asymptoticallyto the k = 0 instabilities in a crossed-over way which we illustrate only by the k = 104 caseIn other words for large k the MAA and MZA instabilites strongly mix with each other

4 Two-dimensional system

We now turn to a 2D system corresponding to the SN example where neutrinos propagatewithin the ldquoexpanding transverse sheetrdquo moving outward The radial motion is parameterizedby our ldquotimerdquo variable whereas ldquospacerdquo is represented by two transverse directions Thisexample corresponds with properly scaled variables micro and λ to the usual treatment of self-induced flavor conversion in SNe except that now we can include small-scale instabilitieswith nonvanishing wavenumber k

41 Single angle (|v| = 1)

We begin with the ldquosingle zenith angle caserdquo meaning that the rescaled neutrino speed withinthe transverse sheet is |v| = 1 and the matter effect can be rotated away Our velocity phasespace is the unit circle described by an angle variable ϕ which we can measure relative tok As the system is initially prepared axially symmetric all vectors k are equivalent mdash theeigenvalues depend only on k = |k| The eigenvalue equation (229) therefore reads

(k cϕ + ω minus Ω)QΩkωϕ = micro

int +infin

int +π

minusπdϕprime (1minus cϕcϕprime minus sϕsϕprime)QΩkωprimeϕprime (41)

where cϕ = cosϕ and sϕ = sinϕ

Proceeding as in our earlier cases we notice that the rhs of equation (41) has theform A1 +Ac cosϕ+As sinϕ ie a superposition of three linearly independent functions onthe interval minusπ le ϕ le +π Therefore the lhs must be of that form as well and we mayuse the eigenfunction ansatz

QΩkωϕ =A1 +Accϕ +Assϕk cϕ + ω minus Ω

ndash 22 ndash

JCAP01(2016)028

λndash

k = 01041

k = 0 102 103 104

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

Figure 7 Footprint of the 1D instabilities (κ gt 10minus2) in the micro-λ-plane for α = 12 and the indicatedvalues of k The homogeneous case (k = 0) is the footprint of the contour plot of figure 6 here on alogarithmic scale The corresponding large-λ asymptotic results are shown in appendix C figure 15

We may insert this form on both sides leading to three linearly independent equationscorresponding to the coefficients of the three functions 1 cosϕ and sinϕ We may write thethree equations in compact form1minus

I1 Ic 0minusIc minusIcc 00 0 minusIss

= 0 (43)

Ia = micro

int +infin

minusinfindω hω

int +π

minusπdϕ

fa(ϕ)

k cϕ + ω minus Ω (44)

ndash 23 ndash

JCAP01(2016)028

Here f1(ϕ) = 1 fc(ϕ) = cosϕ fcc(ϕ) = cos2 ϕ and fss(ϕ) = sin2 ϕ We have used that suchintegrals vanish if they involve a single power of sinϕ because this is anti-symmetric on theintegration interval explaining the zeroes in the matrix in equation (43) We also note thatIss = I1 minus Icc so we need only three different integrals

We first consider homogeneous solutions (k = 0) where the angle integrals in equation (44)can be performed explicitly Using the monochromatic frequency spectrum of equation (228)the eigenvalue equation becomes

1 0 00 minus1

2 00 0 minus1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (45)

These are three independent quadratic equations of the now-familiar form of equation (33)The first line corresponds to the usual bimodal solution the second and third line to twodegenerate multi-azimuth-angle (MAA) solutions which are unstable for negative micro (normalhierarchy) For these modes the instability range is a factor of 2 larger

For the inhomogeneous modes (k gt 0) the required integrals entering the eigenvalue equationare of the form

Ia =micro

int +infin

minusinfindω hωFa

(ω minus Ω

)where Fa(w) =

int +π

minusπdϕ

fa(ϕ)

cosϕ+ w (46)

With our monochromatic spectrum equation (228) we arrive at

Ia =micro

(ω0 minus Ω

)minus αFa

(minusω0 minus Ω

)] (47)

We define the auxiliary function of a complex argument w

s(w) =radicw minus 1

radicw + 1 (48)

which for complex numbers is in general not equal toradicw2 minus 1 We then find

s(w) Fc = 1minus w

s(w) Fcc = minusw +

s(w) Fss = F1 minus Fcc = w minus s(w) (49)

These expressions allow us to write the eigenvalue equations explicitly involving only poly-nomials and square-root expressions

We now have three non-degenerate solutions in contrast to the original 1D exampleof Duan and Shalgar [52] one for positive micro (inverted hierarchy) and two for negative microWe show contour plots of the growth rate κ as a function of micro and k for our usual exampleα = 12 in figure 8 The two quasi-symmetric regions correspond to the two solutions whichcorrespond to those of the 1D case ie the left-right symmetric and anti-symmetric casesalso shown in reference [52] in a similar plot In addition we see a third solution whichis genuinely a result of the spatial 2D geometry with non-vanishing wave-vector k The

ndash 24 ndash

JCAP01(2016)028

-3000 -2000 -1000 0 1000 2000

InteractionStrengthμ

Figure 8 Growth rates for the 2D case with |v| = 1 (ldquosingle zenith anglerdquo) using α = 12 The twoquasi-symmetric regions correspond to the two instabilities which already appear in the 1D case [52]although here the unstable micro range shifts with k34 The third instability is genuinely 2D it has nocounterpart in the ldquocolliding beamrdquo examples and its unstable micro-range shifts linearly with k

eigenfunctions in this case are proportional to sinϕ where ϕ is the angle between k and thevelocity v of a given mode

The eigenvalue equation for the decoupled ss-block of equation (43) the new genuine2D solution is explicitly

(1 + α)ω0 minus (1minus α) Ω + αradicminusk minus ω0 minus Ω

radick minus ω0 minus Ω

minusradicminusk + ω0 minus Ω

radick + ω0 minus Ω = minusk

micro (410)

which is a quartic equation It has unstable solutions for micro lt 0 In analogy to the discussionof the 1D case we can extract the large-k limiting solution with the substitution Ω = Ωminus kand micro = minusk(1minusα)minus

radic4mω0 k (1 + α)(1minus α)3 where m is a dimensionless variable The

detailed factors are not crucial and are the result of some tinkering After the substitutionone expands equation (410) for large k and keeps only the term proportional to the highestpower of k One finds unstable solutions for 0 le m le 1 and the growth rate κω0 =4radicm(1minusm)α(1minusα2) with κmaxω0 = 2α(1minusα2) which is 43 for our example α = 12

The unstable micro-range scales linearly with k Notice that the asymptotic solution obtainsonly for very large k-values because the dominant term in the expansion of equation (410)is proportional to k32 the next one proportional to k and becomes relatively unimportantonly very slowly

The 2times2 block in equation (43) leads to a much more complicated equation but stillconsists of polynomials and square-roots We follow a similar approach and substitute Ω =

Ωminus k and micro = mω140 k34

radic1minus α expand the eigenvalue equation in powers of k and keep

only the largest term leading toradic

2radicminusω0 minus Ω

radicω0 minus Ω = m2

radicω0

(radicminusω0 minus Ωminus α

radicω0 minus Ω

This quartic equation provides the large-k solution which is symmetric for both hierarchiesie symmetric under m rarr minusm Because the second-largest power of k is only a power 14smaller than the largest the asymptotic solution requires extremely large k-values Noticethat in the 1D beam example the unstable range scaled with k12 while here it is k34

ndash 25 ndash

JCAP01(2016)028

We finally turn to the ldquomulti zenith anglerdquo case of transverse velocities within the full 2D diskdescribed by |v| le 1 meaning that multi-angle matter effects are now included We describethe velocity phase space by the speed v = |v| and an angle variable ϕ which we measurerelative to k as before Noting that (1Γv)

intdv = (1π)

int +πminusπ dϕ

int 10 dv v the eigenvalue

equation (229) becomes(12 λv

2 + k v cϕ + ω minus Ω)QΩkωvϕ

= micro

int +infin

int +π

minusπ

dϕprime

0dvprime vprime

[12(vprime2 + v2)minus vvprime (cϕcϕprime + sϕsϕprime)

]QΩkωprimevprimeϕprime (411)

where cϕ = cosϕ and sϕ = sinϕ The rhs as a function of v and ϕ is A0 +A2v2 +Acv cϕ +

Asv sϕ so the eigenfunctions are of the form

QΩkωvϕ =A0 +A2v

2 +Acv cϕ +Asv sϕ12 λv

2 + k v cϕ + ω minus Ω (412)

As usual we insert this form on both sides leading to four linearly independent equationscorresponding to the coefficients of the four functions 1 v2 v cϕ and v sϕ We may write theequations in compact form1minus

3 I15 Ic

1 I13 Ic

2 0minus2Ic

2 minus2Ic4 minus2Icc

3 00 0 0 minus2Iss

= 0 (413)

where we have used thatintdϕprime vanishes for integrals involving an odd power of sinϕprime The

integrals are now

Ian = micro

int +infin

minusinfindω hω

int +π

minusπ

vn fa(ϕ)12 λv

2 + k v cϕ + ω minus Ω (414)

Here f1(ϕ) = 1 fc(ϕ) = cosϕ fcc(ϕ) = cos2 ϕ and fss(ϕ) = sin2 ϕ We also note thatIss

3 = I13 minus Icc

422 Homogeneous mode (k = 0) with matter (λ gt 0)

In the homogeneous case (k = 0) the function cosϕ in the denominator disappears all angleintegrals can be performed analytically and Ic

n = 0 Therefore equation (413) becomes1minus

I3 I5 0 0I1 I3 0 00 0 minusI3 00 0 0 minusI3

= 0 (415)

where the integral expressions after performing the dϕ integration are

In = micro

int +infin

minusinfindω hωKn where Kn =

12 λv

2 + ω minus Ω (416)

ndash 26 ndash

JCAP01(2016)028

The velocity integrals are explicitly

2(ω minus Ω)

) (417a)

[1minus 2(ω minus Ω)

2(ω minus Ω)

)](417b)

2minus 2(ω minus Ω)

(2(ω minus Ω)

2(ω minus Ω)

)] (417c)

The previous ss and cc blocks in equation (413) have now decoupled from the rest and aredegenerate leading to the MAA instability The remaining 2times2 block provides the bimodaland MZA instability In other words we now need to solve

in analogy to reference [45] with a slightly different notation This entire development is verysimilar to the 1D case

In the limit λrarr 0 K1 K3 and K5 approach 12 14 and 16 times 1(ω minus Ω) whichone can also find by setting λ = 0 before doing the v-integrations The matrix can bediagonalized and we find three independent quadratic equations of the form of equation (33)where we need to substitute micro rarr minusmicro4 and micro rarr micro (3 plusmn 2

radic3)12 Following the same steps

as in section 322 the instability ranges for our usual example α = 12 are found to be

127 lt microω0 lt 4328 (419a)

These results are numerically similar to equation (320) for the corresponding 1D-caseIn analogy to the butterfly diagram of the 1D case (figure 6) we show in figure 9 a

contour plot of the growth rate κ in the two-dimensional parameter space of the interactionstrength micro and the effective matter density λ = λ+ εmicro The result looks qualitatively similarto the 1D case Again for micro gt 0 (inverted mass ordering) we obtain the bimodal instabilitywhile for micro lt 0 (normal ordering) we find the MZA and MAA instabilities

423 Inhomogeneous mode (k gt 0) without matter (λ = 0)

Next we consider the relatively simple case of k gt 0 without matter We may write theintegrals of equation (414) in the form

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v cϕ + w(420)

and we have introduced w = (ω minus Ω)k We find explicitly

K11 = w +

radic1minus w

radicminusw(1 + w)radicw

(421a)

radicminuswradic

1minus w2 (1 + 2w2)

3radicw

(421b)

radicminuswradic

1minus w2 (3 + 4w2 + 8w4)

15radicw

(421c)

2minus w2 minus

radicminusw2

radic1minus w2 (421d)

ndash 27 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

Figure 9 Growth rate κ of the 2D instabilities as a function of micro and λ assuming α = 12 Upperpanel the first block in the eigenvalue equation (418) yields the bimodal instability for micro gt 0 (invertedhierarchy) and the multi-zenith-angle (MZA) instability for micro lt 0 (normal hierarchy) Lower panelthe second block in equation (418) provides the multi-azimuth-angle (MAA) instability for micro lt 0This figure is analogous to the corresponding 1D case shown in figure 6

4minus 2w4 +

radicminusw2radic

1minus w2 (1 + 2w2)

3 (421e)

Kcc3 = minusw

2minus w2 minus

radicminusw2

radic1minus w2

) (421f)

Kss3 =

w(3minus 2w2)

radicminusw(1minus w2)32

3radicw

(421g)

Notice that Kss3 +Kcc

3 = K13

With the help of these analytic integrals it is relatively easy to solve the eigenvalueequation numerically We show a contour plot of the growth rate κ in the micro-k-plane in

ndash 28 ndash

JCAP01(2016)028

-1000 -500 0 500

Figure 10 Growth rates for the 2D case with 0 lt |v| = 1 (ldquomulti zenith anglerdquo) using α = 12This figure is analogous to the single-angle case shown in figure 8 but now we have three instabilitiesfor micro lt 0 (normal mass ordering) and the usual bimodal one for micro gt 0 (inverted ordering) For allfour instabilities the unstable micro range scales linearly with k as discussed in the text

figure 10 The 3times3 block in equation (413) provides three different solutions ie onefor micro gt 0 (the usual bimodal solution in inverted mass ordering) and two solutions formicro lt 0 (normal ordering) The 1times1 block provides a further solution for micro lt 0 Figure 10corresponds to figure 8 in the single-angle case In comparison we have one more instabilitynow a total of four of which three are for micro lt 0 (normal mass ordering)

One may also extract the large-k asymptotic behavior (appendix D) For the 3times3 blockin equation (413) one finds that the system is unstable for

micro = ai

radick)

for 0 lt m lt mmaxi where i = 1 3 (422)

ie the unstable micro region scales linearly with k in contrast to the corresponsing 1D caseThe values of the coefficients ai and mmaxi are given in appendix D

For the 1times1 block in equation (413) one finds an instability on a very narrow striparound micro = minus6 k However the maximum growth rate decreases with kminus12 so that in thelimit k rarrinfin this instability disappears and we are left with those arising from the 3times3 block

424 Inhomogeneous mode (k gt 0) with matter (λ gt 0)

As a grand finale we now turn to the most general 2D case with matter (λ gt 0) andinhomogeneities (k gt 0) We need to find the zeroes of the determinant in equation (414)and write the integrals in the form

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

where q = kλ and w = (ωminusΩ)λ These integrals can be performed analytically we provideour results in appendix A

We next determine numerically the instability footprints for non-vanishing wave num-bers k and show the result in figure 11 which looks qualitatively similar to the corresponding

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

Figure 11 Footprint of the 2D instabilities (κ gt 10minus2) in the micro-λ-plane for α = 12 and the indicatedvalues of k The homogeneous case (k = 0) is the footprint of the contour plot of figure 9 here on alogarithmic scale The corresponding large-λ asymptotic results are shown in appendix E figure 16

1D case that was shown in figure 7 For the simpler micro gt 0 half of the plot we show theinstability footprints for k = 0 102 and 103 as indicated in the plot These k gt 0 footprintsfill the space between the k = 0 footprint and the horizontal axis In addition in the upperpanel there are small ldquonosesrdquo of the k gt 0 footprints which slightly extend in the spaceabove the k = 0 footprint but this is a very small effect

For micro lt 0 the situation is more complicated because there are three instabilities Asin the 1D case for large k the footprint connects asymptotically to the k = 0 instabilitiesin a crossed-over way which we illustrate for k = 103 The main novelty of the 2D case isthe appearance of another instability which merges with one of the others when λ k Inother words one of the instabilities somewhat splits into two unstable ranges We also recallthat for very large k the maximum growth rate of one of them decreases and vanishes for

ndash 30 ndash

JCAP01(2016)028

k rarr infin in which case we are back to a total of three instabilities In the unphysical thirdquadrant we notice somewhat pronounced ldquonosesrdquo of the k gt 0 footprints

In all cases the main message is the same as in the 1D case the small-scale instabilitiesfill the space between the k = 0 instability and the horizontal axis whereas the space betweenthe k = 0 instability and the vertical axis remains stable

5 Conclusions

Several recent papers [46 51ndash53] have studied the phenomenon of spatial spontaneous sym-metry breaking in the ldquocolliding beamrdquo model of neutrinos interacting with each other re-fractively One important finding was that for any neutrino density (or in our nomenclaturefor any value of the neutrino-neutrino interaction energy micro) there is some range of spatialwave vectors k where the system is unstable with regard to self-induced flavor conversionAs a consequence it seemed that an interacting neutrino gas would never be stable for anyconditions with potentially far-reaching consequences for SN physics

We have studied similar models but including the multi-angle matter effect We concurwith the previous results in that smaller-scale modes are unstable at larger values of micro for agiven matter density This means that on a ldquofootprint plotrdquo such as figure 11 modes withk gt 0 fill the space between the traditional footprint for k = 0 and the horizontal axis butnot the space towards the vertical axis If we show the instability footprint in a plot likefigure 1 adapted to more physical SN parameters the large-k modes extend the instabilityregion in the direction of larger neutrino density for fixed matter density

Therefore if the instability footprint of the traditional homogeneous (or rather spher-ically symmetric) mode does not intersect with the SN density profile the large-k modesare safe as well In this sense the traditional large-scale mode remains the most sensitivestability probe On the other hand if the physical SN profile of density and neutrino fluxesintersects any instability region instabilities on a large range of spatial scales will occur andone would not expect any simple outcome of the flavor conversion process

Our analysis is based on a linearized stability analysis of a model which we have de-veloped in section 2 We have formulated the problem is such a way that it includes asspecial cases the ldquocolliding beamrdquo examples of the previous literature allows the inclusionof multi-angle effects by a simple modification of phase-space integration and formulates theSN case as a 2D system evolving in time ie the non-trivial dynamical evolution is in the 2Dexpanding sheet of neutrinos as a function of SN distance All cases of the previous literatureare covered in a single simple formulation

In particular our homogeneous multi-angle 2D system corresponds to the usual sce-nario described in the previous literature [45 82] where both the zenith and azimuthalmulti-angle effects are included Note that the usual single zenith angle description of theearly days of the collective oscillation discussions [22] cover only a small part of the parameterspace for example the bimodal instabilities shown in our figure 8 In addition for the firsttime we have have considered a scenario where inhomogeneous modes are included in thedescription of SN neutrino evolution We have found that the homogeneous mode remainsthe dominant source of instability

Still in many ways our investigation is only a mathematical case study that may or maynot apply to a realistic SN Our main simplification is that we assume the flavor content of theSN neutrino stream to be stationary and to evolve only as a function of distance Moreover weassume a uniform boundary condition at the neutrino sphere ie global spherical symmetry

ndash 31 ndash

JCAP01(2016)028

of neutrino emission In other words our case study still contains substantial and nontrivialsimplifying assumptions to reduce the complexity of the full problem Future work will haveto go beyond some of these simplifications to develop a more realistic understanding of whatreally happens to neutrino flavor in the dense SN environment

Acknowledgments

We acknowledge partial support by the Deutsche Forschungsgemeinschaft through Grant NoEXC 153 (Excellence Cluster ldquoUniverserdquo) and by the European Union through the InitialTraining Network ldquoInvisiblesrdquo Grant No PITN-GA-2011-289442 and a Marie Curie Fellow-ship for SC Grant No PIIF-GA-2011-299861 RH acknowledges MPP for hospitality atthe beginning of this project SC and GR acknowledge the Mainz Institute for TheoreticalPhysics (MITP) for hospitality and partial support during the completion of this work

A Analytic integrals

When searching for the complex eigenvalues Ω we need various integrals that can be foundeasily with Wolframrsquos Mathematica There can be issues about the validity of the analyticexpressions in the complex plane so we here give the integrals explicitly

In the 1D case for a non-vanishing k and in the absence of matter effects (λ = 0) weneed integrals of the form

fn(w) =

int +1

minus1dv

v + w (A1)

where w is a complex number We first define two auxiliary functions

L(w) = log

(w + 1

w minus 1

) (A2a)

A(w) = 2 + w[i π sign Im(w)minus 2 arctanh(w)

] (A2b)

The required integrals are found to be

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f2(w) = minus2w + w2L(w) (A3c)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

The actual argument will be of the form w = (ω minus Ω)kFor non-vanishing matter effects (λ 6= 0) and non-vanishing k we need integrals of

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

where w is a complex number and p is real Again we define two auxiliary functions

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

(2minus pradic4w minus p2

)+ arctan

(2 + pradic4w minus p2

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

g1(p w) = minuspB(p w) +K(p w) (A6b)

g2(p w) = 2 +(p2 minus 2w

)B(p w)minus pK(p w) (A6c)

g3(p w) = minus2pminus p(p2 minus 3w

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)+(p4 minus 4p2w + 2w2

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

The actual arguments are going to be p = 2kλ and w = 2(ω minus Ω)λIn the 2D case to solve equation (414) we need the integrals defined in equation (421)

ie integrals of the form

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

where f1(ϕ) = 1 fc(ϕ) = cosϕ fcc(ϕ) = cos2 ϕ and fss(ϕ) = sin2 ϕ These integrals can befound analytically with the help of Mathematica We first define two auxiliary functions

Aqw = arctan

(q2 minus wradicminusw2

)+ arctan

(1minus 2q2 + 2wradic4q2 minus (1 + 2w)2

) (A8a)

Bqw = 2radicminusw2 minus

radic4q2 minus (1 + 2w)2 (A8b)

Our desired integrals are then found to be

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

[minus2radicminusw2 +

(6q2 minus 6w + 1

)Bqw + 4

(3q4 minus 6q2w + 2w2

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

1 +[2radicminusw2 minus

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

minus1minus 4w minus[2radicminusw2 minus

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

where Sw = minusi sign(Imw)

B Frequently encountered eigenvalue equations

Based on our ldquomonochromaticrdquo neutrino spectrum equation (228) with vacuum oscillationfrequencies plusmnω0 and α the number of antineutrinos relative to neutrinos we constantlyencounter eigenvalue equations of the form

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

where micro is an interaction energy In the simplest case F (x) = x this is the traditionaleigenvalue equation for the flavor pendulum if we notice that our micro = micro(1minusα)2 whereas microis the traditional interaction energy We also encounter F (x) =

radicx and F (x) = log(x) To

study these equations we transform them to dimensionless variables by the substitutions

micro = mω01 + α

1minus αand Ω = wω0

1 + α

1minus α (B2)

leading to

(1minus α)(1 + α)minus w

)minus αF

minus(1minus α)(1 + α)minus w

)= 1minus α (B3)

These substitutions allow us to easily take the limit of a ldquosymmetricrdquo neutrino distributionwith ε = 1minus αrarr 0

For the simplest case of the linear function F (x) = x equation (B3) becomes aquadratic equation It has the solution

w = minusmplusmn

radicminus(2minusm)m+

(1minus α1 + α

It has a nonvanishing imaginary part in the range

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

We denote with K1(αm) the function which is the positive imaginary part of w In thepresent case it simply is

K1(αm) =

radic(2minusm)mminus

(1minus α1 + α

As a function of m it is a semi-circle with center at m = 1 and radius 2radicα(1 + α) ie

Kmax1 =

2radicα

1 + α (B7)

This function is shown for several values of α in the top panel of figure 12 In the limit ofequal neutrino and antineutrino densities ie for α rarr 1 the imaginary part is K(1m) =radic

(2minusm)m This limiting result can be found from equation (B3) directly by substitutingε = 1minusα expanding the equation in powers of ε and keeping only the lowest power leadingto the equation 2m+ 2mw + w2 = 0

To compare with section 312 notice that micro = micro(1 minus α)2 = mω0(1 + α)(1 minus α)Using this relationship between micro and m as well as the one between Ω and w reproduces theprevious results In particular for αrarr 1 we find the physical growth rate κ =

radic(2microminus ω0)ω0

which grows without limit for micro rarr infin It is interesting that in terms of the scaled variablem one obtains as a limiting results for α rarr 1 a semi-circle for K(1m) as a function ofm So in principle one could study all of our problems in the α rarr 1 limit and yet obtainrepresentative results for α lt 1 ie one could essentially eliminate the annoying parameterα and still obtain meaningful results for the asymmetric case

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

Figure 12 The functions K(αm) as defined in the text for the cases F (x) = xradicx and log(x) as

indicated for α = 1 03 01 003 and 001 in each case from outside in

The next more complicated case is the square-root function F (x) =radicx for which

equation (B3) takes on the form(2m

minus α(

= 1minus α (B8)

It can be transformed to a quartic equation but the results are too cumbersome to deal withand not particularly informative Again we can expand this equation in powers of 1minusα andfind for the limit α rarr 1 the cubic equation 2w3 +m(1 + 2w)2 = 0 The imaginary part ofthe solution is

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

radic81minus 48mminus 27

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

This function is shown in the second panel of figure 12 and looks almost like a semi-circlebut is not quite one It is is nonzero for 0 lt m lt 2716 = 16875 and takes on its maximum

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

show K12(αm) also for several other value of α The curves always begin at m = 0 iethere is no lower threshold in this case

We finally turn to the logarithmic case F (x) = log(x) natural logarithm always un-derstood for which equation (B3) becomes

)minus α log

)= 1minus α (B10)

There is no general analytic solution but again we can consider the αrarr 1 expansion wherewe find 1 = w[1 + log(minusw2m)] The solution is w(m) = 1W (minuse2m) where e is Eulerrsquosnumber and W (z) is the Lambert W -function ie W (z) is the solution of z = W eW InMathematica it is implemented as W (z) = ProductLog[z] The positive imaginary partof our w(m) ie Klog(1m) = Im[minus1W (minuse2m)] is nonzero for 0 lt m lt e22 = 369453and is shown in the bottom panel of figure 12 Again it looks deceivingly like a semi-circlebut is not exactly one Its maximum occurs at m = 176684 and Kmax

log = 0724611

We usually consider α = 12 as our main example In this case the eigenvalue equationcan be solved analytically with the solution

wα=12 =eminus 3mplusmn

radic3m(3mminus 4e)

3e (B11)

The imaginary part is nonzero for 0 lt m lt 4e3 has its maximum at m = 2e3 and takeson a maximum value of 23

C Asymptotic solutions for 1D and λrarrinfin

We can derive analytic asymptotic solutions for the 1D case with matter ie the large-λcontinuation of the contour plot figure 6 We begin with the bimodal and MZA instability forλ gt 0 and consider the first block of the eigenvalue equation (322) It is of the form 1+C1micro+C2micro

2 = 0 where the coefficients C1 and C2 depend on α ω0 Ω and λ We have evaluated theintegrals according to the explicit transcendental functions given in equation (A6) Inspiredby numerical solutions we assume that both the real and imaginary parts of the solutions Ωremain of order ω0 and do not become large as λ rarr infin an assumption that later bears outto be consistent with the solutions Therefore we may expand C1 and C2 in powers of λminus1

and find that the dominant terms are C1 prop λminus1 and C2 prop λminus32 In terms of a dimensionlessinteraction strength micro of order unity we write

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

where the exact coefficient was chosen for later convenience The lowest-order term in C2micro2

no longer depends on λ whereas the lowest-order term in C1micro prop λminus14 and slowly becomessmall as λrarrinfin To lowest order in λminus1 the eigenvalue equation is found to beradic

ω0 minus Ωminus α

radicω0

minusω0 minus Ω=

1minus αmicro2

This is an example of the type of equations that we always encounter in this context andwhich are discussed in appendix B

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

Figure 13 Growth rate κ of the bimodal and MAA instabilities using α = 12 The interactionstrength is scaled according to equation (C1) The blue curves show the asymptotic behavior forλrarrinfin the other curves are for the indicated λ values

The asymptotic solution derives from the term quadratic in micro and thus remains un-changed under microrarr minusmicro ie it applies to both hierarchies We show the asymptotic solutionas a blue curve in figure 13 on a linear and logarithmic scale We also show the growth ratesfor λ = 102 104 and 106 where the solution is not symmetric under micro rarr minusmicro because thelinear term in micro kicks in We have already noted that one needs very large λ values to obtainthe asymptotic solution because the second-largest term only scales with λminus14 relative tothe dominant term The asymptotic behavior is achieved for much smaller λ values if micro gt 0The growth rate vanishes completely above a certain |micro| value but obtains nonzero valuesotherwise ie there is no lower micro threshold However for micro 05 the growth rate is a steeppower-law of micro and can be taken to be effectively zero

For our usual example α = 12 we find numerically that the maximum growth rateoccurs for micro = 1494 Therefore we find that

micro = plusmn 4911ω140 λ34 (C3)

gives us the locus of the maximum growth rate in the micro-λ plane for the bimodal and MAAsolutions The maximum value of micro before the growth rate becomes zero is 17724 On thesmall-micro side the growth rate drops below κ lt 1100 our usual criterion at micro = 03478Therefore the footprint of the instability is the region between the lines micro = 1143λ34 and5826λ34 where both micro and λ are given in units of the vacuum oscillation frequency ω0This footprint is shown in the first quadrant (upper right) of figure 15 The correspondingfootprint in the second quadrant (upper left) is also shown

We next turn to the MZA solution which exists only in inverted hierarchy (micro lt 0) andwe consider the second block in equation (322) If we use micro = minusλ[2(1minusα)] the leading termscancel leaving us with a leading term of order λminus12 To obtain the lowest-order equationwe introduce another dimensionless parameter micro and write

micro =minusλ

2 (1minus α)minus micro π

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

where of course the detailed coefficients in the second term are chosen for later convenienceOne then finds a quadratic equation with solutions

1 + α2 minus 2micro2 (1 + α2)

1minus α2plusmn i 4α

1minus α2

radicmicro2(1minus micro2) (C5)

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

Rescaled Interaction Strength μ

GrowthRate

102103λ = infin

Figure 14 Asymptotic growth rate κ for the MZA instability using α = 12 The interactionstrength is scaled according to equation (C4) The blue curve shows the asymptotic behavior forλrarrinfin according to equation (C5) the other curves are for λ = 102 and 103 from outside in

Notice that these solutions require 0 le micro le 1 and we have always assumed 0 le α le 1 Theimaginary part as a function of micro2 has the familiar semi-circular shape In figure 14 weshow it as a function of micro (blue curve) and we also show the full solution for λ = 103 and102 The asymptotic solution is quite good for relatively small λ values In contrast to theother solutions on a logarithmic scale the unstable range becomes very narrow as λrarrinfin

The maximum growth rate obtains for micro = 1radic

2 Therefore for our usual exampleα = 12 we find that for the MZA solution

micro = minusλminus πradic

3ω0λ2 (C6)

gives us the locus of the maximum growth rate in the micro-λ plane The growth rate becomesexactly zero for micro le 0 and micro ge 1 so the footprint (see upper-left quadrant in figure 14) is

delimited by the curves micro = minusλ and micro = minusλ minus πradic

3λω0 The width of the footprint scaleswith

radicλ ie on a logarithmic scale it becomes very narrow for large λ

For λ lt 0 the above approach does not lead to unstable solutions Numerically weobserve that for λrarr minusinfin the real part of the solutions approaches Re(Ω)rarr λ2 ie a largenegative number Therefore to be able to expand the equation we express Ω = λ2 + wω0

and seek self-consistent solutions with the dimensionless eigenvalue w of order unity Afterexpansion for λrarr minusinfin the asymptotic eigenvalue equations are

log(1minus w)minus α log(minus1minus w)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

(3minus micro)micro (C8)

where e is Eulerrsquos number and as always the logarithm is with base e This equation is oneexample for the type discussed in appendix B

For our usual example α = 12 we can solve this equation analytically with the ex-plicit result

wα=12 =2minus ea plusmn i

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

It has a nonzero imaginary part for minusinfin lt a lt log(8) = 20794 although it becomesexponentially small for a minus1 The maximum imaginary part obtains for a = log(4) =13863 and the maximum is 2 Therefore the maximum growth rate obtains for

microor A = minus 3 + micro2

A = minus log(4)minus 2 + log

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

Therefore we have altogether three solutions corresponding to the three instabilities withmaximum growth rates on the locus in the micro-λ plane given by

micro = minus2λ6

3A+radic

3(A+ 2)(3Aminus 2)rarr minus2λ

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3(A+ 2)(3Aminus 2)

2 (Aminus 1)rarr 6λ (C12c)

where the limiting behavior is understood for A rarr infin Because λ rarr minusinfin the first solutioncorresponds to positive micro and thus to the bimodal solution the second and third solutionsare the MAA and MZA instabilities respectively

To draw the footprints in the lower quadrants of figure 15 we notice that κ = 0 fora gt log(8) and on the other side κ lt 1100 for a lt log(4 minus

radic3999950) = minus990348

Therefore the asymptotic footprints are limited by

a1 = log(8) and a2 = log(4minusradic

3999950) (C13)

from which the limiting curves are extracted by solving equation (C8) for micro Once more wenote that two of the footprints are ldquowiderdquo and nearly symmetric between microrarr minusmicro whereasthe third instability has a very narrow footprint

D Asymptotic solutions for 2D with λ = 0 and krarrinfin

We are looking for the large-k solutions of the 2D case without matter (λ = 0) We needto find the zeroes of the determinant of the matrix in equation (413) We first look at the3times 3 block and calculate it according to the explicit integrals that we have found Next wesubstitute the variables as ω = 1 α = 12 Ω = minusk + x and

micro = a (k +mradick) (D1)

where a is a coefficient to be determined and overall the substitution for micro is an educatedguess Except for the choice α = 12 everything is still completely general The unknownfrequency to be found is x Its imaginary part is the growth rate which we are lookingfor The parameter m is an effective interaction strength because it gives us micro in thisparameterised form

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

Figure 15 Footprint of the 1D instabilities in the micro-λ plane for k = 0 (homogeneous mode) andα = 12 as explained in the text The colored regions derive from a numerical solution where theblue footprints correspond to the 2times2 block in equation (322) the red solutions to the 1times1 blockThe grey regions show the asymptotic solutions in the large-λ limit derived in this appendix

Next we expand the determinant as a power series for large k and find to lowest non-trivial order

det(3times3 block) =2880minus 480 aminus 424 a2 minus 11 a3

minus 3 i

[radic2a2(32 + a)

(2radicxminus 1minus

radicx+ 1

)] 1radick

+O(1k) (D2)

For term proportional to 1radick to dominate we demand the first term to vanish giving us

three possible values for a from the requirement 2880 minus 480 a minus 424 a2 minus 11 a3 = 0 The

ndash 40 ndash

JCAP01(2016)028

explicit results are quite complicated expressions Numerically one finds

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

In other words we have three asymptotic solutions where one is for positive micro and two fornegative micro as expected

If we now imagine that a is one of these solutions the first term in the determinantvanishes and in the second term we can substitute a3 = (2880minus 480 aminus 424 a2)11 to removethe a3 term In anticipation of the result we further introduce the quantity

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

which for our three possible a values are numerically

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

Then we are left with the equivalent of the determinant equationradic

6m = immax

radicx+ 1

) (D10)

It has the explicit solutions

3minus 10m2

3m2max

plusmn8radicm2(m2 minusm2

3m2max

The solution has an imaginary part for 0 lt m lt mmax Therefore the large-k footprint ofthe three instabilities is limited by the lines

micro = ai k and micro = ai

(k +mmaxi

radick) (D12)

For micro-values between these lines the system is unstableFinally we turn to the 1times1 block in equation (413) We proceed with the same substi-

tutions except formicro = a (k + b) (D13)

where for the moment we leave open what b is supposed to mean Expanding the 1times1 blockdeterminant in powers of large k we here find

det(1times1 block) =6 + a

6(minus9 + b+ 3x)

+ i2radic

[2(xminus 1)32 minus (x+ 1)32

k32+O(1k2) (D14)

Again we can get rid of the first term this time by setting a = minus6 ie the footprint of thisinstability is for negative micro The remaining equation is

det(1times1 block) = (9minus bminus 3x)1

kminus i 4radic

2[2(xminus 1)32 minus (x+ 1)32

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

The leading term does not provide an imaginary solution In other words for very large k wedo not have an instability If we keep both the leading and next to leading term we finallyneed to solve the equation

(9minus bminus 3x)radick = i 4

radic2[2(xminus 1)32 minus (x+ 1)32

] (D16)

Solving this equation actually leads to an asymptotic solution where the growth rate existsfor a range of b-values However the maximum growth rate decreases with 1

radick Therefore

we have overall four instabilities but for k rarrinfin the one from the single block disappears

E Asymptotic solutions for 2D with k = 0 and λrarrinfin

We can derive asymptotic solutions for the 2D case with matter ie the large-λ solutionsof the eigenvalue equation (415) corresponding to the two equations (418) We begin withthe 2times2 block and λ rarr +infin As in the 1D case we assume that Ω remains of order ω0 anassumption which is confirmed by the results We express the interaction strength in termsof a dimensionless parameter micro in the form

micro =micro

1minus αλ (E1)

To lowest order in λminus1 the eigenvalue equation is using w = Ωω0

1minus α= a where a = log

)minus 2(microminus 1)2

micro2 (E2)

This result is identical with equation (C7) but with a different expression for a To drawthe asymptotic footprints we simply need to solve for micro using the limiting a-values givenin equation (C13) The result is shown in figure 16 as grey shaded regions in the upperpanels to be compared with the blue regions which derive from a numerical solution of thefull eigenvalue equations

Next we turn to the 1times1 block for the limit λ rarr +infin and express the interactionstrength in the form

micro = minus λ+ microω0 log(λ2ω0)

1minus α (E3)

With micro = 0 the eigenvalue equation is identically fulfilled to lowest order in λminus1 ie to lowestorder unstable solutions require micro = minusλ(1 minus α) This simple behavior indeed correspondsto the very ldquothinrdquo footprint shown in red in the upper left panel of figure 16 Includingmicro 6= 0 leads to an approximate eigenvalue equation which is not very simple and does notlead to simple asymptotic solutions Expressing micro in terms of micro as in equation (E3) we cannumerially find the growth rate κ as a function of micro as shown in figure 17 It is clear thatthe instability footprint in the logarithmic figure 16 will be very narrow We also notice thatthe maximum growth rate decreases with increasing λ (For all of the other instabilities andfor α = 12 the maximum growth rate κmax = 2ω0)

For the next cases we turn to the limit λrarr minusinfin In this limit we write Ω = λ2 +wω0

in analogy to the 1D case In the λrarr minusinfin limit the eigenvalue is characterized by w valuesof order unity We also write the interaction strength again in the form of equation (E1)The limiting eigenvalue equation is the same as in equation (E2) but now with

a = log

(minus λ

(microminus 4) microor a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

Figure 17 Growth rate κ for the instability deriving from the 1times1 block in equation (418) usingα = 12 and solving the full eigenvalue equation The interaction strength is scaled according toequation (E3) The curves are for the indicated values of λ

ndash 43 ndash

JCAP01(2016)028

where the first expression applies to the 2times2 block the second to the 1times1 block of theeigenvalue matrix As before to draw the asymptotic footprints we solve for micro using thelimiting a-values given in equation (C13) The result is shown in figure 16 as grey shadedregions in the lower panels to be compared with the blue and red regions which derive froma numerical solution of the full equations

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[80] T Melson H-T Janka and A Marek Neutrino-driven supernova of a low-mass iron-coreprogenitor boosted by three-dimensional turbulent convection Astrophys J 801 (2015) L24[arXiv150101961] [INSPIRE]

[81] T Melson H-T Janka R Bollig F Hanke A Marek and B Muller Neutrino-drivenExplosion of a 20 Solar-mass Star in Three Dimensions Enabled by Strange-quark Contributionsto Neutrino-nucleon Scattering Astrophys J 808 (2015) L42 [arXiv150407631] [INSPIRE]

[82] S Chakraborty G Raffelt H-T Janka and B Muller Supernova deleptonization asymmetryImpact on self-induced flavor conversion Phys Rev D 92 (2015) 105002 [arXiv14120670][INSPIRE]

ndash 48 ndash

Introduction

Equations of motion

Setting up the system

Large-distance approximation

Single angle vs multi angle

Two-flavor system

Mass ordering

Linearization

Homogeneous neutrino and electron densities

Spatial Fourier transform

Oscillation eigenmodes

Monochromatic and isotropic neutrino distribution

One-dimensional system

Single (angle v=+-1)

Eigenvalue equation

Homogeneous mode (k=0)

Inhomogeneous modes (kgt0)

Multi-angle effects (0lt= vlt=1)

Eigenvalue equation

Homogeneous mode (k=0) without matter effects (bar(lambda)=0)

Inhomogeneous modes (kgt0) without matter effects (bar(lambda)=0)

Including matter(bar(lambda)=0)


Inhomogeneous mode (kgt0)

Two-dimensional system

Single angle (|v|=1)

Eigenvalue equation




Eigenvalue equation

Homogeneous mode (k=0) with matter (bar(lambda)gt0)

Inhomogeneous mode (kgt0) without matter (bar(lambda)=0)

Inhomogeneous mode (kgt0) with matter (bar(lambda)gt0)

Conclusions

Analytic integrals

Frequently encountered eigenvalue equations

Asymptotic solutions for 1D and (lambda) to infty

Asymptotic solutions for 2D with bar(lambda)=0 and k to infty

Asymptotic solutions for 2D with k=0 and bar(lambda) to infty

JCAP01(2016)028

Contents

1 Introduction 1

2 Equations of motion 421 Setting up the system 422 Large-distance approximation 523 Single angle vs multi angle 624 Two-flavor system 725 Mass ordering 926 Linearization 927 Homogeneous neutrino and electron densities 1028 Spatial Fourier transform 1129 Oscillation eigenmodes 11210 Monochromatic and isotropic neutrino distribution 11

3 One-dimensional system 1231 Single angle (v = plusmn1) 12

32 Multi-angle effects (0 le v le 1) 14321 Eigenvalue equation 14322 Homogeneous mode (k = 0) without matter effects (λ = 0) 15323 Inhomogeneous modes (k gt 0) without matter effects (λ = 0) 18

33 Including matter(λ 6= 0) 19331 Homogeneous mode (k = 0) 19332 Inhomogeneous mode (k gt 0) 22

4 Two-dimensional system 2241 Single angle (|v| = 1) 22

42 Multi-angle effects (0 le v le 1) 26421 Eigenvalue equation 26422 Homogeneous mode (k = 0) with matter (λ gt 0) 26423 Inhomogeneous mode (k gt 0) without matter (λ = 0) 27424 Inhomogeneous mode (k gt 0) with matter (λ gt 0) 29

5 Conclusions 31

A Analytic integrals 32

B Frequently encountered eigenvalue equations 33

C Asymptotic solutions for 1D and λrarrinfin 36

ndash i ndash

JCAP01(2016)028

1 Introduction

ndash 1 ndash

JCAP01(2016)028

radic2GFnνe(Rr)

ndash 2 ndash

JCAP01(2016)028

50 100 200 500 1000

10610-1100101102103104

Radius HkmL

Μ Hkm-1L

SNDensity

k = 0102103

2GFne(Rr)2

ndash 3 ndash

JCAP01(2016)028

2E+radic

intdΓprime

(v minus vprime)2

2trEprimevprime

] (22)

intdΓprime =

2(v minus vprime)2

ndash 4 ndash

JCAP01(2016)028

where vz =radic

2) or explicitly

2E+radic

)+radic

intdΓprime

(β minus βprime)2

radic1minus β2

2β2 + 1

ndash 5 ndash

JCAP01(2016)028

β2max

2E+radic

)+radic

2GFβ2max

intdΓprime

(v minus vprime)2

ndash 6 ndash

JCAP01(2016)028

langM2

rang+radic

2GFN`β2max

2+radic

2GFβ2max

intdΓprime

(v minus vprime)2

ω =∆m2

2E= 063 kmminus1

(10 MeV

) (29)

rangrarr ω

2 00 minus1

) (210)

ndash 7 ndash

JCAP01(2016)028

radic2GFN`β

2rarr λv2

2 00 minus1

) (211)

λ =radic

2GFne(r)R2

1012 g cmminus3

r2 (212)

micro =radic

2GFnνe(r)R2

4times1052 ergs

10 MeV

〈Eνe〉

(30 km

)2 (Rr

txωv =Tr(txωv)

2+nνe2

Gtxωv (214)

ndash 8 ndash

JCAP01(2016)028

Htxωv =(ω + λx

12v2)(+1

2 00 minus1

)+ micro

intdΓprime

(v minus vprime)2

Gtxωprimevprime

2 (216)

25 Mass ordering

26 Linearization

(g GGlowast minusg

)(218)

ndash 9 ndash

JCAP01(2016)028

[ω + λx

12v2 + micro

intdΓprime 1

]Gtxωv

minus gtxωv[micro

intdΓprime 1

] (219b)

intdΓ gωv ε1 =

ω + 12λv2 + 1

Stxωv =Gtxωvgωv

intdΓprime 1

ndash 10 ndash

JCAP01(2016)028

iStkωv =(ω + 1

intdωprime

intdvprime 1

intinfin0 dω

intdv gωv = 1 In

)QΩkωv = micro

intdωprime

intdvprime 1

gωv = hω fv (227)

ndash 11 ndash

JCAP01(2016)028

)QΩkωv = micro

minus Ω

= 0 (31)

)micro2 = 1 (32)

ndash 12 ndash

JCAP01(2016)028

[(1 + α)ω0 + (1minus α)Ω

]= 0 (33)

2plusmn

radic[ω0 +

(1minus α)micro

(1 +radicα)2

ltmicro

(1minusradicα)2

κmax =2radicα

1minus αω0 (36)

(1minus α)2 (37)

1minus 2radicα

1 + αlt

microκmax

lt 1 +2radicα

1 + α (38)

radicω0k

minusω0 minus Ω

)m2 = 1 (39)

ndash 13 ndash

JCAP01(2016)028

ω0= minusε

4plusmnradic

4 (310)

micro2κmax

=4 (1 + α)

ndash 14 ndash

JCAP01(2016)028

int +infin

int +1

12 λ v

2 + k v + ω minus Ω(313)

A0 +A1v +A2v2 =

int +infin

int +1

A0 +A1vprime +A2v

prime2

12 λ v

I2 I3 I4

I0 I1 I2

= 0 (315)

In =micro

int +infin

minusinfindω hω

int +1

minus1dv

12 λ v

2 + k v + ω minus Ω (316)

In =micro

2(1 + n)

ω0 + Ω+

ω0 minus Ω

2(1 + n)

ω20 minus Ω2

ndash 15 ndash

JCAP01(2016)028

13 0 1

50 minus2

3 01 0 1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (318)

microrarr micro

30times

radic5 gt 0

minus10 lt 0

5minus 3radic

5 lt 0

176 lt microω0 lt 5974 (320a)

ndash 16 ndash

JCAP01(2016)028

GrowthRateκ

k = 0 λ = 0 nv = 1

GrowthRateκ

k = 0 λ = 0 nv = 2

-500 -400 -300 -200 -100 0 1000

GrowthRateκ

k = 0 λ = 0 nv = 10

ndash 17 ndash

JCAP01(2016)028

-800 -600 -400 -200 00

GrowthRateκ

5101520

-150 -100 -50 0 50 100 1500

GrowthRateκ

1010 2020

-3000 -2500 -2000 -1500 -1000 -500 00

GrowthRateκ

255075100

-400 -200 0 200 4000

GrowthRateκ

2525 5050 7575 100100

ndash 18 ndash

JCAP01(2016)028

I2 0 I4

0 minus2I2 0I0 0 I2

= 0 (321)

ndash 19 ndash

JCAP01(2016)028

GrowthRateκ

k = 25 nv = 2

GrowthRateκ

k = 25 nv = 5

GrowthRateκ

k = 25 nv = 10

-1200-1000 -800 -600 -400 -200 0 2000

GrowthRateκ

k = 25 nv = 20

ndash 20 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

ndash 21 ndash

JCAP01(2016)028

int +infin

int +π

ndash 22 ndash

JCAP01(2016)028

λndash

k = 01041

k = 0 102 103 104

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

= 0 (43)

Ia = micro

int +infin

minusinfindω hω

int +π

minusπdϕ

fa(ϕ)

ndash 23 ndash

JCAP01(2016)028

1 0 00 minus1

2 00 0 minus1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (45)

Ia =micro

int +infin

minusinfindω hωFa

(ω minus Ω

)where Fa(w) =

int +π

minusπdϕ

fa(ϕ)

cosϕ+ w (46)

Ia =micro

(ω0 minus Ω

)minus αFa

(minusω0 minus Ω

)] (47)

radicw + 1 (48)

s(w) Fc = 1minus w

s(w) Fcc = minusw +

ndash 24 ndash

JCAP01(2016)028

-3000 -2000 -1000 0 1000 2000

micro (410)

radicω0

radicω0 minus Ω

ndash 25 ndash

JCAP01(2016)028

intdv = (1π)

int +πminusπ dϕ

= micro

int +infin

int +π

minusπ

dϕprime

0dvprime vprime

QΩkωvϕ =A0 +A2v

2 + k v cϕ + ω minus Ω (412)

3 I15 Ic

1 I13 Ic

2 0minus2Ic

3 00 0 0 minus2Iss

= 0 (413)

integrals are now

Ian = micro

int +infin

minusinfindω hω

int +π

minusπ

vn fa(ϕ)12 λv

2 + k v cϕ + ω minus Ω (414)

3 = I13 minus Icc

= 0 (415)

In = micro

int +infin

12 λv

2 + ω minus Ω (416)

ndash 26 ndash

JCAP01(2016)028

2(ω minus Ω)

) (417a)

2(ω minus Ω)

)](417b)

(2(ω minus Ω)

2(ω minus Ω)

)] (417c)

127 lt microω0 lt 4328 (419a)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v cϕ + w(420)

K11 = w +

radic1minus w

(421a)

radicminuswradic

1minus w2 (1 + 2w2)

3radicw

(421b)

radicminuswradic

1minus w2 (3 + 4w2 + 8w4)

15radicw

(421c)

2minus w2 minus

radicminusw2

ndash 27 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

4minus 2w4 +

radicminusw2radic

1minus w2 (1 + 2w2)

3 (421e)

Kcc3 = minusw

2minus w2 minus

radicminusw2

radic1minus w2

) (421f)

Kss3 =

w(3minus 2w2)

3radicw

(421g)

3 = K13

ndash 28 ndash

JCAP01(2016)028

-1000 -500 0 500

micro = ai

radick)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

1 Introduction

ndash 1 ndash

JCAP01(2016)028

radic2GFnνe(Rr)

ndash 2 ndash

JCAP01(2016)028

50 100 200 500 1000

10610-1100101102103104

Radius HkmL

Μ Hkm-1L

SNDensity

k = 0102103

2GFne(Rr)2

ndash 3 ndash

JCAP01(2016)028

2E+radic

intdΓprime

(v minus vprime)2

2trEprimevprime

] (22)

intdΓprime =

2(v minus vprime)2

ndash 4 ndash

JCAP01(2016)028

where vz =radic

2) or explicitly

2E+radic

)+radic

intdΓprime

(β minus βprime)2

radic1minus β2

2β2 + 1

ndash 5 ndash

JCAP01(2016)028

β2max

2E+radic

)+radic

2GFβ2max

intdΓprime

(v minus vprime)2

ndash 6 ndash

JCAP01(2016)028

langM2

rang+radic

2GFN`β2max

2+radic

2GFβ2max

intdΓprime

(v minus vprime)2

ω =∆m2

2E= 063 kmminus1

(10 MeV

) (29)

rangrarr ω

2 00 minus1

) (210)

ndash 7 ndash

JCAP01(2016)028

radic2GFN`β

2rarr λv2

2 00 minus1

) (211)

λ =radic

2GFne(r)R2

1012 g cmminus3

r2 (212)

micro =radic

2GFnνe(r)R2

4times1052 ergs

10 MeV

〈Eνe〉

(30 km

)2 (Rr

txωv =Tr(txωv)

2+nνe2

Gtxωv (214)

ndash 8 ndash

JCAP01(2016)028

Htxωv =(ω + λx

12v2)(+1

2 00 minus1

)+ micro

intdΓprime

(v minus vprime)2

Gtxωprimevprime

2 (216)

25 Mass ordering

26 Linearization

(g GGlowast minusg

)(218)

ndash 9 ndash

JCAP01(2016)028

[ω + λx

12v2 + micro

intdΓprime 1

]Gtxωv

minus gtxωv[micro

intdΓprime 1

] (219b)

intdΓ gωv ε1 =

ω + 12λv2 + 1

Stxωv =Gtxωvgωv

intdΓprime 1

ndash 10 ndash

JCAP01(2016)028

iStkωv =(ω + 1

intdωprime

intdvprime 1

intinfin0 dω

intdv gωv = 1 In

)QΩkωv = micro

intdωprime

intdvprime 1

gωv = hω fv (227)

ndash 11 ndash

JCAP01(2016)028

)QΩkωv = micro

minus Ω

= 0 (31)

)micro2 = 1 (32)

ndash 12 ndash

JCAP01(2016)028

[(1 + α)ω0 + (1minus α)Ω

]= 0 (33)

2plusmn

radic[ω0 +

(1minus α)micro

(1 +radicα)2

ltmicro

(1minusradicα)2

κmax =2radicα

1minus αω0 (36)

(1minus α)2 (37)

1minus 2radicα

1 + αlt

microκmax

lt 1 +2radicα

1 + α (38)

radicω0k

minusω0 minus Ω

)m2 = 1 (39)

ndash 13 ndash

JCAP01(2016)028

ω0= minusε

4plusmnradic

4 (310)

micro2κmax

=4 (1 + α)

ndash 14 ndash

JCAP01(2016)028

int +infin

int +1

12 λ v

2 + k v + ω minus Ω(313)

A0 +A1v +A2v2 =

int +infin

int +1

A0 +A1vprime +A2v

prime2

12 λ v

I2 I3 I4

I0 I1 I2

= 0 (315)

In =micro

int +infin

minusinfindω hω

int +1

minus1dv

12 λ v

2 + k v + ω minus Ω (316)

In =micro

2(1 + n)

ω0 + Ω+

ω0 minus Ω

2(1 + n)

ω20 minus Ω2

ndash 15 ndash

JCAP01(2016)028

13 0 1

50 minus2

3 01 0 1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (318)

microrarr micro

30times

radic5 gt 0

minus10 lt 0

5minus 3radic

5 lt 0

176 lt microω0 lt 5974 (320a)

ndash 16 ndash

JCAP01(2016)028

GrowthRateκ

k = 0 λ = 0 nv = 1

GrowthRateκ

k = 0 λ = 0 nv = 2

-500 -400 -300 -200 -100 0 1000

GrowthRateκ

k = 0 λ = 0 nv = 10

ndash 17 ndash

JCAP01(2016)028

-800 -600 -400 -200 00

GrowthRateκ

5101520

-150 -100 -50 0 50 100 1500

GrowthRateκ

1010 2020

-3000 -2500 -2000 -1500 -1000 -500 00

GrowthRateκ

255075100

-400 -200 0 200 4000

GrowthRateκ

2525 5050 7575 100100

ndash 18 ndash

JCAP01(2016)028

I2 0 I4

0 minus2I2 0I0 0 I2

= 0 (321)

ndash 19 ndash

JCAP01(2016)028

GrowthRateκ

k = 25 nv = 2

GrowthRateκ

k = 25 nv = 5

GrowthRateκ

k = 25 nv = 10

-1200-1000 -800 -600 -400 -200 0 2000

GrowthRateκ

k = 25 nv = 20

ndash 20 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

ndash 21 ndash

JCAP01(2016)028

int +infin

int +π

ndash 22 ndash

JCAP01(2016)028

λndash

k = 01041

k = 0 102 103 104

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

= 0 (43)

Ia = micro

int +infin

minusinfindω hω

int +π

minusπdϕ

fa(ϕ)

ndash 23 ndash

JCAP01(2016)028

1 0 00 minus1

2 00 0 minus1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (45)

Ia =micro

int +infin

minusinfindω hωFa

(ω minus Ω

)where Fa(w) =

int +π

minusπdϕ

fa(ϕ)

cosϕ+ w (46)

Ia =micro

(ω0 minus Ω

)minus αFa

(minusω0 minus Ω

)] (47)

radicw + 1 (48)

s(w) Fc = 1minus w

s(w) Fcc = minusw +

ndash 24 ndash

JCAP01(2016)028

-3000 -2000 -1000 0 1000 2000

micro (410)

radicω0

radicω0 minus Ω

ndash 25 ndash

JCAP01(2016)028

intdv = (1π)

int +πminusπ dϕ

= micro

int +infin

int +π

minusπ

dϕprime

0dvprime vprime

QΩkωvϕ =A0 +A2v

2 + k v cϕ + ω minus Ω (412)

3 I15 Ic

1 I13 Ic

2 0minus2Ic

3 00 0 0 minus2Iss

= 0 (413)

integrals are now

Ian = micro

int +infin

minusinfindω hω

int +π

minusπ

vn fa(ϕ)12 λv

2 + k v cϕ + ω minus Ω (414)

3 = I13 minus Icc

= 0 (415)

In = micro

int +infin

12 λv

2 + ω minus Ω (416)

ndash 26 ndash

JCAP01(2016)028

2(ω minus Ω)

) (417a)

2(ω minus Ω)

)](417b)

(2(ω minus Ω)

2(ω minus Ω)

)] (417c)

127 lt microω0 lt 4328 (419a)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v cϕ + w(420)

K11 = w +

radic1minus w

(421a)

radicminuswradic

1minus w2 (1 + 2w2)

3radicw

(421b)

radicminuswradic

1minus w2 (3 + 4w2 + 8w4)

15radicw

(421c)

2minus w2 minus

radicminusw2

ndash 27 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

4minus 2w4 +

radicminusw2radic

1minus w2 (1 + 2w2)

3 (421e)

Kcc3 = minusw

2minus w2 minus

radicminusw2

radic1minus w2

) (421f)

Kss3 =

w(3minus 2w2)

3radicw

(421g)

3 = K13

ndash 28 ndash

JCAP01(2016)028

-1000 -500 0 500

micro = ai

radick)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

radic2GFnνe(Rr)

ndash 2 ndash

JCAP01(2016)028

50 100 200 500 1000

10610-1100101102103104

Radius HkmL

Μ Hkm-1L

SNDensity

k = 0102103

2GFne(Rr)2

ndash 3 ndash

JCAP01(2016)028

2E+radic

intdΓprime

(v minus vprime)2

2trEprimevprime

] (22)

intdΓprime =

2(v minus vprime)2

ndash 4 ndash

JCAP01(2016)028

where vz =radic

2) or explicitly

2E+radic

)+radic

intdΓprime

(β minus βprime)2

radic1minus β2

2β2 + 1

ndash 5 ndash

JCAP01(2016)028

β2max

2E+radic

)+radic

2GFβ2max

intdΓprime

(v minus vprime)2

ndash 6 ndash

JCAP01(2016)028

langM2

rang+radic

2GFN`β2max

2+radic

2GFβ2max

intdΓprime

(v minus vprime)2

ω =∆m2

2E= 063 kmminus1

(10 MeV

) (29)

rangrarr ω

2 00 minus1

) (210)

ndash 7 ndash

JCAP01(2016)028

radic2GFN`β

2rarr λv2

2 00 minus1

) (211)

λ =radic

2GFne(r)R2

1012 g cmminus3

r2 (212)

micro =radic

2GFnνe(r)R2

4times1052 ergs

10 MeV

〈Eνe〉

(30 km

)2 (Rr

txωv =Tr(txωv)

2+nνe2

Gtxωv (214)

ndash 8 ndash

JCAP01(2016)028

Htxωv =(ω + λx

12v2)(+1

2 00 minus1

)+ micro

intdΓprime

(v minus vprime)2

Gtxωprimevprime

2 (216)

25 Mass ordering

26 Linearization

(g GGlowast minusg

)(218)

ndash 9 ndash

JCAP01(2016)028

[ω + λx

12v2 + micro

intdΓprime 1

]Gtxωv

minus gtxωv[micro

intdΓprime 1

] (219b)

intdΓ gωv ε1 =

ω + 12λv2 + 1

Stxωv =Gtxωvgωv

intdΓprime 1

ndash 10 ndash

JCAP01(2016)028

iStkωv =(ω + 1

intdωprime

intdvprime 1

intinfin0 dω

intdv gωv = 1 In

)QΩkωv = micro

intdωprime

intdvprime 1

gωv = hω fv (227)

ndash 11 ndash

JCAP01(2016)028

)QΩkωv = micro

minus Ω

= 0 (31)

)micro2 = 1 (32)

ndash 12 ndash

JCAP01(2016)028

[(1 + α)ω0 + (1minus α)Ω

]= 0 (33)

2plusmn

radic[ω0 +

(1minus α)micro

(1 +radicα)2

ltmicro

(1minusradicα)2

κmax =2radicα

1minus αω0 (36)

(1minus α)2 (37)

1minus 2radicα

1 + αlt

microκmax

lt 1 +2radicα

1 + α (38)

radicω0k

minusω0 minus Ω

)m2 = 1 (39)

ndash 13 ndash

JCAP01(2016)028

ω0= minusε

4plusmnradic

4 (310)

micro2κmax

=4 (1 + α)

ndash 14 ndash

JCAP01(2016)028

int +infin

int +1

12 λ v

2 + k v + ω minus Ω(313)

A0 +A1v +A2v2 =

int +infin

int +1

A0 +A1vprime +A2v

prime2

12 λ v

I2 I3 I4

I0 I1 I2

= 0 (315)

In =micro

int +infin

minusinfindω hω

int +1

minus1dv

12 λ v

2 + k v + ω minus Ω (316)

In =micro

2(1 + n)

ω0 + Ω+

ω0 minus Ω

2(1 + n)

ω20 minus Ω2

ndash 15 ndash

JCAP01(2016)028

13 0 1

50 minus2

3 01 0 1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (318)

microrarr micro

30times

radic5 gt 0

minus10 lt 0

5minus 3radic

5 lt 0

176 lt microω0 lt 5974 (320a)

ndash 16 ndash

JCAP01(2016)028

GrowthRateκ

k = 0 λ = 0 nv = 1

GrowthRateκ

k = 0 λ = 0 nv = 2

-500 -400 -300 -200 -100 0 1000

GrowthRateκ

k = 0 λ = 0 nv = 10

ndash 17 ndash

JCAP01(2016)028

-800 -600 -400 -200 00

GrowthRateκ

5101520

-150 -100 -50 0 50 100 1500

GrowthRateκ

1010 2020

-3000 -2500 -2000 -1500 -1000 -500 00

GrowthRateκ

255075100

-400 -200 0 200 4000

GrowthRateκ

2525 5050 7575 100100

ndash 18 ndash

JCAP01(2016)028

I2 0 I4

0 minus2I2 0I0 0 I2

= 0 (321)

ndash 19 ndash

JCAP01(2016)028

GrowthRateκ

k = 25 nv = 2

GrowthRateκ

k = 25 nv = 5

GrowthRateκ

k = 25 nv = 10

-1200-1000 -800 -600 -400 -200 0 2000

GrowthRateκ

k = 25 nv = 20

ndash 20 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

ndash 21 ndash

JCAP01(2016)028

int +infin

int +π

ndash 22 ndash

JCAP01(2016)028

λndash

k = 01041

k = 0 102 103 104

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

= 0 (43)

Ia = micro

int +infin

minusinfindω hω

int +π

minusπdϕ

fa(ϕ)

ndash 23 ndash

JCAP01(2016)028

1 0 00 minus1

2 00 0 minus1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (45)

Ia =micro

int +infin

minusinfindω hωFa

(ω minus Ω

)where Fa(w) =

int +π

minusπdϕ

fa(ϕ)

cosϕ+ w (46)

Ia =micro

(ω0 minus Ω

)minus αFa

(minusω0 minus Ω

)] (47)

radicw + 1 (48)

s(w) Fc = 1minus w

s(w) Fcc = minusw +

ndash 24 ndash

JCAP01(2016)028

-3000 -2000 -1000 0 1000 2000

micro (410)

radicω0

radicω0 minus Ω

ndash 25 ndash

JCAP01(2016)028

intdv = (1π)

int +πminusπ dϕ

= micro

int +infin

int +π

minusπ

dϕprime

0dvprime vprime

QΩkωvϕ =A0 +A2v

2 + k v cϕ + ω minus Ω (412)

3 I15 Ic

1 I13 Ic

2 0minus2Ic

3 00 0 0 minus2Iss

= 0 (413)

integrals are now

Ian = micro

int +infin

minusinfindω hω

int +π

minusπ

vn fa(ϕ)12 λv

2 + k v cϕ + ω minus Ω (414)

3 = I13 minus Icc

= 0 (415)

In = micro

int +infin

12 λv

2 + ω minus Ω (416)

ndash 26 ndash

JCAP01(2016)028

2(ω minus Ω)

) (417a)

2(ω minus Ω)

)](417b)

(2(ω minus Ω)

2(ω minus Ω)

)] (417c)

127 lt microω0 lt 4328 (419a)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v cϕ + w(420)

K11 = w +

radic1minus w

(421a)

radicminuswradic

1minus w2 (1 + 2w2)

3radicw

(421b)

radicminuswradic

1minus w2 (3 + 4w2 + 8w4)

15radicw

(421c)

2minus w2 minus

radicminusw2

ndash 27 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

4minus 2w4 +

radicminusw2radic

1minus w2 (1 + 2w2)

3 (421e)

Kcc3 = minusw

2minus w2 minus

radicminusw2

radic1minus w2

) (421f)

Kss3 =

w(3minus 2w2)

3radicw

(421g)

3 = K13

ndash 28 ndash

JCAP01(2016)028

-1000 -500 0 500

micro = ai

radick)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

50 100 200 500 1000

10610-1100101102103104

Radius HkmL

Μ Hkm-1L

SNDensity

k = 0102103

2GFne(Rr)2

ndash 3 ndash

JCAP01(2016)028

2E+radic

intdΓprime

(v minus vprime)2

2trEprimevprime

] (22)

intdΓprime =

2(v minus vprime)2

ndash 4 ndash

JCAP01(2016)028

where vz =radic

2) or explicitly

2E+radic

)+radic

intdΓprime

(β minus βprime)2

radic1minus β2

2β2 + 1

ndash 5 ndash

JCAP01(2016)028

β2max

2E+radic

)+radic

2GFβ2max

intdΓprime

(v minus vprime)2

ndash 6 ndash

JCAP01(2016)028

langM2

rang+radic

2GFN`β2max

2+radic

2GFβ2max

intdΓprime

(v minus vprime)2

ω =∆m2

2E= 063 kmminus1

(10 MeV

) (29)

rangrarr ω

2 00 minus1

) (210)

ndash 7 ndash

JCAP01(2016)028

radic2GFN`β

2rarr λv2

2 00 minus1

) (211)

λ =radic

2GFne(r)R2

1012 g cmminus3

r2 (212)

micro =radic

2GFnνe(r)R2

4times1052 ergs

10 MeV

〈Eνe〉

(30 km

)2 (Rr

txωv =Tr(txωv)

2+nνe2

Gtxωv (214)

ndash 8 ndash

JCAP01(2016)028

Htxωv =(ω + λx

12v2)(+1

2 00 minus1

)+ micro

intdΓprime

(v minus vprime)2

Gtxωprimevprime

2 (216)

25 Mass ordering

26 Linearization

(g GGlowast minusg

)(218)

ndash 9 ndash

JCAP01(2016)028

[ω + λx

12v2 + micro

intdΓprime 1

]Gtxωv

minus gtxωv[micro

intdΓprime 1

] (219b)

intdΓ gωv ε1 =

ω + 12λv2 + 1

Stxωv =Gtxωvgωv

intdΓprime 1

ndash 10 ndash

JCAP01(2016)028

iStkωv =(ω + 1

intdωprime

intdvprime 1

intinfin0 dω

intdv gωv = 1 In

)QΩkωv = micro

intdωprime

intdvprime 1

gωv = hω fv (227)

ndash 11 ndash

JCAP01(2016)028

)QΩkωv = micro

minus Ω

= 0 (31)

)micro2 = 1 (32)

ndash 12 ndash

JCAP01(2016)028

[(1 + α)ω0 + (1minus α)Ω

]= 0 (33)

2plusmn

radic[ω0 +

(1minus α)micro

(1 +radicα)2

ltmicro

(1minusradicα)2

κmax =2radicα

1minus αω0 (36)

(1minus α)2 (37)

1minus 2radicα

1 + αlt

microκmax

lt 1 +2radicα

1 + α (38)

radicω0k

minusω0 minus Ω

)m2 = 1 (39)

ndash 13 ndash

JCAP01(2016)028

ω0= minusε

4plusmnradic

4 (310)

micro2κmax

=4 (1 + α)

ndash 14 ndash

JCAP01(2016)028

int +infin

int +1

12 λ v

2 + k v + ω minus Ω(313)

A0 +A1v +A2v2 =

int +infin

int +1

A0 +A1vprime +A2v

prime2

12 λ v

I2 I3 I4

I0 I1 I2

= 0 (315)

In =micro

int +infin

minusinfindω hω

int +1

minus1dv

12 λ v

2 + k v + ω minus Ω (316)

In =micro

2(1 + n)

ω0 + Ω+

ω0 minus Ω

2(1 + n)

ω20 minus Ω2

ndash 15 ndash

JCAP01(2016)028

13 0 1

50 minus2

3 01 0 1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (318)

microrarr micro

30times

radic5 gt 0

minus10 lt 0

5minus 3radic

5 lt 0

176 lt microω0 lt 5974 (320a)

ndash 16 ndash

JCAP01(2016)028

GrowthRateκ

k = 0 λ = 0 nv = 1

GrowthRateκ

k = 0 λ = 0 nv = 2

-500 -400 -300 -200 -100 0 1000

GrowthRateκ

k = 0 λ = 0 nv = 10

ndash 17 ndash

JCAP01(2016)028

-800 -600 -400 -200 00

GrowthRateκ

5101520

-150 -100 -50 0 50 100 1500

GrowthRateκ

1010 2020

-3000 -2500 -2000 -1500 -1000 -500 00

GrowthRateκ

255075100

-400 -200 0 200 4000

GrowthRateκ

2525 5050 7575 100100

ndash 18 ndash

JCAP01(2016)028

I2 0 I4

0 minus2I2 0I0 0 I2

= 0 (321)

ndash 19 ndash

JCAP01(2016)028

GrowthRateκ

k = 25 nv = 2

GrowthRateκ

k = 25 nv = 5

GrowthRateκ

k = 25 nv = 10

-1200-1000 -800 -600 -400 -200 0 2000

GrowthRateκ

k = 25 nv = 20

ndash 20 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

ndash 21 ndash

JCAP01(2016)028

int +infin

int +π

ndash 22 ndash

JCAP01(2016)028

λndash

k = 01041

k = 0 102 103 104

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

= 0 (43)

Ia = micro

int +infin

minusinfindω hω

int +π

minusπdϕ

fa(ϕ)

ndash 23 ndash

JCAP01(2016)028

1 0 00 minus1

2 00 0 minus1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (45)

Ia =micro

int +infin

minusinfindω hωFa

(ω minus Ω

)where Fa(w) =

int +π

minusπdϕ

fa(ϕ)

cosϕ+ w (46)

Ia =micro

(ω0 minus Ω

)minus αFa

(minusω0 minus Ω

)] (47)

radicw + 1 (48)

s(w) Fc = 1minus w

s(w) Fcc = minusw +

ndash 24 ndash

JCAP01(2016)028

-3000 -2000 -1000 0 1000 2000

micro (410)

radicω0

radicω0 minus Ω

ndash 25 ndash

JCAP01(2016)028

intdv = (1π)

int +πminusπ dϕ

= micro

int +infin

int +π

minusπ

dϕprime

0dvprime vprime

QΩkωvϕ =A0 +A2v

2 + k v cϕ + ω minus Ω (412)

3 I15 Ic

1 I13 Ic

2 0minus2Ic

3 00 0 0 minus2Iss

= 0 (413)

integrals are now

Ian = micro

int +infin

minusinfindω hω

int +π

minusπ

vn fa(ϕ)12 λv

2 + k v cϕ + ω minus Ω (414)

3 = I13 minus Icc

= 0 (415)

In = micro

int +infin

12 λv

2 + ω minus Ω (416)

ndash 26 ndash

JCAP01(2016)028

2(ω minus Ω)

) (417a)

2(ω minus Ω)

)](417b)

(2(ω minus Ω)

2(ω minus Ω)

)] (417c)

127 lt microω0 lt 4328 (419a)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v cϕ + w(420)

K11 = w +

radic1minus w

(421a)

radicminuswradic

1minus w2 (1 + 2w2)

3radicw

(421b)

radicminuswradic

1minus w2 (3 + 4w2 + 8w4)

15radicw

(421c)

2minus w2 minus

radicminusw2

ndash 27 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

4minus 2w4 +

radicminusw2radic

1minus w2 (1 + 2w2)

3 (421e)

Kcc3 = minusw

2minus w2 minus

radicminusw2

radic1minus w2

) (421f)

Kss3 =

w(3minus 2w2)

3radicw

(421g)

3 = K13

ndash 28 ndash

JCAP01(2016)028

-1000 -500 0 500

micro = ai

radick)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

2E+radic

intdΓprime

(v minus vprime)2

2trEprimevprime

] (22)

intdΓprime =

2(v minus vprime)2

ndash 4 ndash

JCAP01(2016)028

where vz =radic

2) or explicitly

2E+radic

)+radic

intdΓprime

(β minus βprime)2

radic1minus β2

2β2 + 1

ndash 5 ndash

JCAP01(2016)028

β2max

2E+radic

)+radic

2GFβ2max

intdΓprime

(v minus vprime)2

ndash 6 ndash

JCAP01(2016)028

langM2

rang+radic

2GFN`β2max

2+radic

2GFβ2max

intdΓprime

(v minus vprime)2

ω =∆m2

2E= 063 kmminus1

(10 MeV

) (29)

rangrarr ω

2 00 minus1

) (210)

ndash 7 ndash

JCAP01(2016)028

radic2GFN`β

2rarr λv2

2 00 minus1

) (211)

λ =radic

2GFne(r)R2

1012 g cmminus3

r2 (212)

micro =radic

2GFnνe(r)R2

4times1052 ergs

10 MeV

〈Eνe〉

(30 km

)2 (Rr

txωv =Tr(txωv)

2+nνe2

Gtxωv (214)

ndash 8 ndash

JCAP01(2016)028

Htxωv =(ω + λx

12v2)(+1

2 00 minus1

)+ micro

intdΓprime

(v minus vprime)2

Gtxωprimevprime

2 (216)

25 Mass ordering

26 Linearization

(g GGlowast minusg

)(218)

ndash 9 ndash

JCAP01(2016)028

[ω + λx

12v2 + micro

intdΓprime 1

]Gtxωv

minus gtxωv[micro

intdΓprime 1

] (219b)

intdΓ gωv ε1 =

ω + 12λv2 + 1

Stxωv =Gtxωvgωv

intdΓprime 1

ndash 10 ndash

JCAP01(2016)028

iStkωv =(ω + 1

intdωprime

intdvprime 1

intinfin0 dω

intdv gωv = 1 In

)QΩkωv = micro

intdωprime

intdvprime 1

gωv = hω fv (227)

ndash 11 ndash

JCAP01(2016)028

)QΩkωv = micro

minus Ω

= 0 (31)

)micro2 = 1 (32)

ndash 12 ndash

JCAP01(2016)028

[(1 + α)ω0 + (1minus α)Ω

]= 0 (33)

2plusmn

radic[ω0 +

(1minus α)micro

(1 +radicα)2

ltmicro

(1minusradicα)2

κmax =2radicα

1minus αω0 (36)

(1minus α)2 (37)

1minus 2radicα

1 + αlt

microκmax

lt 1 +2radicα

1 + α (38)

radicω0k

minusω0 minus Ω

)m2 = 1 (39)

ndash 13 ndash

JCAP01(2016)028

ω0= minusε

4plusmnradic

4 (310)

micro2κmax

=4 (1 + α)

ndash 14 ndash

JCAP01(2016)028

int +infin

int +1

12 λ v

2 + k v + ω minus Ω(313)

A0 +A1v +A2v2 =

int +infin

int +1

A0 +A1vprime +A2v

prime2

12 λ v

I2 I3 I4

I0 I1 I2

= 0 (315)

In =micro

int +infin

minusinfindω hω

int +1

minus1dv

12 λ v

2 + k v + ω minus Ω (316)

In =micro

2(1 + n)

ω0 + Ω+

ω0 minus Ω

2(1 + n)

ω20 minus Ω2

ndash 15 ndash

JCAP01(2016)028

13 0 1

50 minus2

3 01 0 1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (318)

microrarr micro

30times

radic5 gt 0

minus10 lt 0

5minus 3radic

5 lt 0

176 lt microω0 lt 5974 (320a)

ndash 16 ndash

JCAP01(2016)028

GrowthRateκ

k = 0 λ = 0 nv = 1

GrowthRateκ

k = 0 λ = 0 nv = 2

-500 -400 -300 -200 -100 0 1000

GrowthRateκ

k = 0 λ = 0 nv = 10

ndash 17 ndash

JCAP01(2016)028

-800 -600 -400 -200 00

GrowthRateκ

5101520

-150 -100 -50 0 50 100 1500

GrowthRateκ

1010 2020

-3000 -2500 -2000 -1500 -1000 -500 00

GrowthRateκ

255075100

-400 -200 0 200 4000

GrowthRateκ

2525 5050 7575 100100

ndash 18 ndash

JCAP01(2016)028

I2 0 I4

0 minus2I2 0I0 0 I2

= 0 (321)

ndash 19 ndash

JCAP01(2016)028

GrowthRateκ

k = 25 nv = 2

GrowthRateκ

k = 25 nv = 5

GrowthRateκ

k = 25 nv = 10

-1200-1000 -800 -600 -400 -200 0 2000

GrowthRateκ

k = 25 nv = 20

ndash 20 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

ndash 21 ndash

JCAP01(2016)028

int +infin

int +π

ndash 22 ndash

JCAP01(2016)028

λndash

k = 01041

k = 0 102 103 104

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

= 0 (43)

Ia = micro

int +infin

minusinfindω hω

int +π

minusπdϕ

fa(ϕ)

ndash 23 ndash

JCAP01(2016)028

1 0 00 minus1

2 00 0 minus1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (45)

Ia =micro

int +infin

minusinfindω hωFa

(ω minus Ω

)where Fa(w) =

int +π

minusπdϕ

fa(ϕ)

cosϕ+ w (46)

Ia =micro

(ω0 minus Ω

)minus αFa

(minusω0 minus Ω

)] (47)

radicw + 1 (48)

s(w) Fc = 1minus w

s(w) Fcc = minusw +

ndash 24 ndash

JCAP01(2016)028

-3000 -2000 -1000 0 1000 2000

micro (410)

radicω0

radicω0 minus Ω

ndash 25 ndash

JCAP01(2016)028

intdv = (1π)

int +πminusπ dϕ

= micro

int +infin

int +π

minusπ

dϕprime

0dvprime vprime

QΩkωvϕ =A0 +A2v

2 + k v cϕ + ω minus Ω (412)

3 I15 Ic

1 I13 Ic

2 0minus2Ic

3 00 0 0 minus2Iss

= 0 (413)

integrals are now

Ian = micro

int +infin

minusinfindω hω

int +π

minusπ

vn fa(ϕ)12 λv

2 + k v cϕ + ω minus Ω (414)

3 = I13 minus Icc

= 0 (415)

In = micro

int +infin

12 λv

2 + ω minus Ω (416)

ndash 26 ndash

JCAP01(2016)028

2(ω minus Ω)

) (417a)

2(ω minus Ω)

)](417b)

(2(ω minus Ω)

2(ω minus Ω)

)] (417c)

127 lt microω0 lt 4328 (419a)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v cϕ + w(420)

K11 = w +

radic1minus w

(421a)

radicminuswradic

1minus w2 (1 + 2w2)

3radicw

(421b)

radicminuswradic

1minus w2 (3 + 4w2 + 8w4)

15radicw

(421c)

2minus w2 minus

radicminusw2

ndash 27 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

4minus 2w4 +

radicminusw2radic

1minus w2 (1 + 2w2)

3 (421e)

Kcc3 = minusw

2minus w2 minus

radicminusw2

radic1minus w2

) (421f)

Kss3 =

w(3minus 2w2)

3radicw

(421g)

3 = K13

ndash 28 ndash

JCAP01(2016)028

-1000 -500 0 500

micro = ai

radick)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

where vz =radic

2) or explicitly

2E+radic

)+radic

intdΓprime

(β minus βprime)2

radic1minus β2

2β2 + 1

ndash 5 ndash

JCAP01(2016)028

β2max

2E+radic

)+radic

2GFβ2max

intdΓprime

(v minus vprime)2

ndash 6 ndash

JCAP01(2016)028

langM2

rang+radic

2GFN`β2max

2+radic

2GFβ2max

intdΓprime

(v minus vprime)2

ω =∆m2

2E= 063 kmminus1

(10 MeV

) (29)

rangrarr ω

2 00 minus1

) (210)

ndash 7 ndash

JCAP01(2016)028

radic2GFN`β

2rarr λv2

2 00 minus1

) (211)

λ =radic

2GFne(r)R2

1012 g cmminus3

r2 (212)

micro =radic

2GFnνe(r)R2

4times1052 ergs

10 MeV

〈Eνe〉

(30 km

)2 (Rr

txωv =Tr(txωv)

2+nνe2

Gtxωv (214)

ndash 8 ndash

JCAP01(2016)028

Htxωv =(ω + λx

12v2)(+1

2 00 minus1

)+ micro

intdΓprime

(v minus vprime)2

Gtxωprimevprime

2 (216)

25 Mass ordering

26 Linearization

(g GGlowast minusg

)(218)

ndash 9 ndash

JCAP01(2016)028

[ω + λx

12v2 + micro

intdΓprime 1

]Gtxωv

minus gtxωv[micro

intdΓprime 1

] (219b)

intdΓ gωv ε1 =

ω + 12λv2 + 1

Stxωv =Gtxωvgωv

intdΓprime 1

ndash 10 ndash

JCAP01(2016)028

iStkωv =(ω + 1

intdωprime

intdvprime 1

intinfin0 dω

intdv gωv = 1 In

)QΩkωv = micro

intdωprime

intdvprime 1

gωv = hω fv (227)

ndash 11 ndash

JCAP01(2016)028

)QΩkωv = micro

minus Ω

= 0 (31)

)micro2 = 1 (32)

ndash 12 ndash

JCAP01(2016)028

[(1 + α)ω0 + (1minus α)Ω

]= 0 (33)

2plusmn

radic[ω0 +

(1minus α)micro

(1 +radicα)2

ltmicro

(1minusradicα)2

κmax =2radicα

1minus αω0 (36)

(1minus α)2 (37)

1minus 2radicα

1 + αlt

microκmax

lt 1 +2radicα

1 + α (38)

radicω0k

minusω0 minus Ω

)m2 = 1 (39)

ndash 13 ndash

JCAP01(2016)028

ω0= minusε

4plusmnradic

4 (310)

micro2κmax

=4 (1 + α)

ndash 14 ndash

JCAP01(2016)028

int +infin

int +1

12 λ v

2 + k v + ω minus Ω(313)

A0 +A1v +A2v2 =

int +infin

int +1

A0 +A1vprime +A2v

prime2

12 λ v

I2 I3 I4

I0 I1 I2

= 0 (315)

In =micro

int +infin

minusinfindω hω

int +1

minus1dv

12 λ v

2 + k v + ω minus Ω (316)

In =micro

2(1 + n)

ω0 + Ω+

ω0 minus Ω

2(1 + n)

ω20 minus Ω2

ndash 15 ndash

JCAP01(2016)028

13 0 1

50 minus2

3 01 0 1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (318)

microrarr micro

30times

radic5 gt 0

minus10 lt 0

5minus 3radic

5 lt 0

176 lt microω0 lt 5974 (320a)

ndash 16 ndash

JCAP01(2016)028

GrowthRateκ

k = 0 λ = 0 nv = 1

GrowthRateκ

k = 0 λ = 0 nv = 2

-500 -400 -300 -200 -100 0 1000

GrowthRateκ

k = 0 λ = 0 nv = 10

ndash 17 ndash

JCAP01(2016)028

-800 -600 -400 -200 00

GrowthRateκ

5101520

-150 -100 -50 0 50 100 1500

GrowthRateκ

1010 2020

-3000 -2500 -2000 -1500 -1000 -500 00

GrowthRateκ

255075100

-400 -200 0 200 4000

GrowthRateκ

2525 5050 7575 100100

ndash 18 ndash

JCAP01(2016)028

I2 0 I4

0 minus2I2 0I0 0 I2

= 0 (321)

ndash 19 ndash

JCAP01(2016)028

GrowthRateκ

k = 25 nv = 2

GrowthRateκ

k = 25 nv = 5

GrowthRateκ

k = 25 nv = 10

-1200-1000 -800 -600 -400 -200 0 2000

GrowthRateκ

k = 25 nv = 20

ndash 20 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

ndash 21 ndash

JCAP01(2016)028

int +infin

int +π

ndash 22 ndash

JCAP01(2016)028

λndash

k = 01041

k = 0 102 103 104

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

= 0 (43)

Ia = micro

int +infin

minusinfindω hω

int +π

minusπdϕ

fa(ϕ)

ndash 23 ndash

JCAP01(2016)028

1 0 00 minus1

2 00 0 minus1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (45)

Ia =micro

int +infin

minusinfindω hωFa

(ω minus Ω

)where Fa(w) =

int +π

minusπdϕ

fa(ϕ)

cosϕ+ w (46)

Ia =micro

(ω0 minus Ω

)minus αFa

(minusω0 minus Ω

)] (47)

radicw + 1 (48)

s(w) Fc = 1minus w

s(w) Fcc = minusw +

ndash 24 ndash

JCAP01(2016)028

-3000 -2000 -1000 0 1000 2000

micro (410)

radicω0

radicω0 minus Ω

ndash 25 ndash

JCAP01(2016)028

intdv = (1π)

int +πminusπ dϕ

= micro

int +infin

int +π

minusπ

dϕprime

0dvprime vprime

QΩkωvϕ =A0 +A2v

2 + k v cϕ + ω minus Ω (412)

3 I15 Ic

1 I13 Ic

2 0minus2Ic

3 00 0 0 minus2Iss

= 0 (413)

integrals are now

Ian = micro

int +infin

minusinfindω hω

int +π

minusπ

vn fa(ϕ)12 λv

2 + k v cϕ + ω minus Ω (414)

3 = I13 minus Icc

= 0 (415)

In = micro

int +infin

12 λv

2 + ω minus Ω (416)

ndash 26 ndash

JCAP01(2016)028

2(ω minus Ω)

) (417a)

2(ω minus Ω)

)](417b)

(2(ω minus Ω)

2(ω minus Ω)

)] (417c)

127 lt microω0 lt 4328 (419a)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v cϕ + w(420)

K11 = w +

radic1minus w

(421a)

radicminuswradic

1minus w2 (1 + 2w2)

3radicw

(421b)

radicminuswradic

1minus w2 (3 + 4w2 + 8w4)

15radicw

(421c)

2minus w2 minus

radicminusw2

ndash 27 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

4minus 2w4 +

radicminusw2radic

1minus w2 (1 + 2w2)

3 (421e)

Kcc3 = minusw

2minus w2 minus

radicminusw2

radic1minus w2

) (421f)

Kss3 =

w(3minus 2w2)

3radicw

(421g)

3 = K13

ndash 28 ndash

JCAP01(2016)028

-1000 -500 0 500

micro = ai

radick)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

β2max

2E+radic

)+radic

2GFβ2max

intdΓprime

(v minus vprime)2

ndash 6 ndash

JCAP01(2016)028

langM2

rang+radic

2GFN`β2max

2+radic

2GFβ2max

intdΓprime

(v minus vprime)2

ω =∆m2

2E= 063 kmminus1

(10 MeV

) (29)

rangrarr ω

2 00 minus1

) (210)

ndash 7 ndash

JCAP01(2016)028

radic2GFN`β

2rarr λv2

2 00 minus1

) (211)

λ =radic

2GFne(r)R2

1012 g cmminus3

r2 (212)

micro =radic

2GFnνe(r)R2

4times1052 ergs

10 MeV

〈Eνe〉

(30 km

)2 (Rr

txωv =Tr(txωv)

2+nνe2

Gtxωv (214)

ndash 8 ndash

JCAP01(2016)028

Htxωv =(ω + λx

12v2)(+1

2 00 minus1

)+ micro

intdΓprime

(v minus vprime)2

Gtxωprimevprime

2 (216)

25 Mass ordering

26 Linearization

(g GGlowast minusg

)(218)

ndash 9 ndash

JCAP01(2016)028

[ω + λx

12v2 + micro

intdΓprime 1

]Gtxωv

minus gtxωv[micro

intdΓprime 1

] (219b)

intdΓ gωv ε1 =

ω + 12λv2 + 1

Stxωv =Gtxωvgωv

intdΓprime 1

ndash 10 ndash

JCAP01(2016)028

iStkωv =(ω + 1

intdωprime

intdvprime 1

intinfin0 dω

intdv gωv = 1 In

)QΩkωv = micro

intdωprime

intdvprime 1

gωv = hω fv (227)

ndash 11 ndash

JCAP01(2016)028

)QΩkωv = micro

minus Ω

= 0 (31)

)micro2 = 1 (32)

ndash 12 ndash

JCAP01(2016)028

[(1 + α)ω0 + (1minus α)Ω

]= 0 (33)

2plusmn

radic[ω0 +

(1minus α)micro

(1 +radicα)2

ltmicro

(1minusradicα)2

κmax =2radicα

1minus αω0 (36)

(1minus α)2 (37)

1minus 2radicα

1 + αlt

microκmax

lt 1 +2radicα

1 + α (38)

radicω0k

minusω0 minus Ω

)m2 = 1 (39)

ndash 13 ndash

JCAP01(2016)028

ω0= minusε

4plusmnradic

4 (310)

micro2κmax

=4 (1 + α)

ndash 14 ndash

JCAP01(2016)028

int +infin

int +1

12 λ v

2 + k v + ω minus Ω(313)

A0 +A1v +A2v2 =

int +infin

int +1

A0 +A1vprime +A2v

prime2

12 λ v

I2 I3 I4

I0 I1 I2

= 0 (315)

In =micro

int +infin

minusinfindω hω

int +1

minus1dv

12 λ v

2 + k v + ω minus Ω (316)

In =micro

2(1 + n)

ω0 + Ω+

ω0 minus Ω

2(1 + n)

ω20 minus Ω2

ndash 15 ndash

JCAP01(2016)028

13 0 1

50 minus2

3 01 0 1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (318)

microrarr micro

30times

radic5 gt 0

minus10 lt 0

5minus 3radic

5 lt 0

176 lt microω0 lt 5974 (320a)

ndash 16 ndash

JCAP01(2016)028

GrowthRateκ

k = 0 λ = 0 nv = 1

GrowthRateκ

k = 0 λ = 0 nv = 2

-500 -400 -300 -200 -100 0 1000

GrowthRateκ

k = 0 λ = 0 nv = 10

ndash 17 ndash

JCAP01(2016)028

-800 -600 -400 -200 00

GrowthRateκ

5101520

-150 -100 -50 0 50 100 1500

GrowthRateκ

1010 2020

-3000 -2500 -2000 -1500 -1000 -500 00

GrowthRateκ

255075100

-400 -200 0 200 4000

GrowthRateκ

2525 5050 7575 100100

ndash 18 ndash

JCAP01(2016)028

I2 0 I4

0 minus2I2 0I0 0 I2

= 0 (321)

ndash 19 ndash

JCAP01(2016)028

GrowthRateκ

k = 25 nv = 2

GrowthRateκ

k = 25 nv = 5

GrowthRateκ

k = 25 nv = 10

-1200-1000 -800 -600 -400 -200 0 2000

GrowthRateκ

k = 25 nv = 20

ndash 20 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

ndash 21 ndash

JCAP01(2016)028

int +infin

int +π

ndash 22 ndash

JCAP01(2016)028

λndash

k = 01041

k = 0 102 103 104

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

= 0 (43)

Ia = micro

int +infin

minusinfindω hω

int +π

minusπdϕ

fa(ϕ)

ndash 23 ndash

JCAP01(2016)028

1 0 00 minus1

2 00 0 minus1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (45)

Ia =micro

int +infin

minusinfindω hωFa

(ω minus Ω

)where Fa(w) =

int +π

minusπdϕ

fa(ϕ)

cosϕ+ w (46)

Ia =micro

(ω0 minus Ω

)minus αFa

(minusω0 minus Ω

)] (47)

radicw + 1 (48)

s(w) Fc = 1minus w

s(w) Fcc = minusw +

ndash 24 ndash

JCAP01(2016)028

-3000 -2000 -1000 0 1000 2000

micro (410)

radicω0

radicω0 minus Ω

ndash 25 ndash

JCAP01(2016)028

intdv = (1π)

int +πminusπ dϕ

= micro

int +infin

int +π

minusπ

dϕprime

0dvprime vprime

QΩkωvϕ =A0 +A2v

2 + k v cϕ + ω minus Ω (412)

3 I15 Ic

1 I13 Ic

2 0minus2Ic

3 00 0 0 minus2Iss

= 0 (413)

integrals are now

Ian = micro

int +infin

minusinfindω hω

int +π

minusπ

vn fa(ϕ)12 λv

2 + k v cϕ + ω minus Ω (414)

3 = I13 minus Icc

= 0 (415)

In = micro

int +infin

12 λv

2 + ω minus Ω (416)

ndash 26 ndash

JCAP01(2016)028

2(ω minus Ω)

) (417a)

2(ω minus Ω)

)](417b)

(2(ω minus Ω)

2(ω minus Ω)

)] (417c)

127 lt microω0 lt 4328 (419a)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v cϕ + w(420)

K11 = w +

radic1minus w

(421a)

radicminuswradic

1minus w2 (1 + 2w2)

3radicw

(421b)

radicminuswradic

1minus w2 (3 + 4w2 + 8w4)

15radicw

(421c)

2minus w2 minus

radicminusw2

ndash 27 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

4minus 2w4 +

radicminusw2radic

1minus w2 (1 + 2w2)

3 (421e)

Kcc3 = minusw

2minus w2 minus

radicminusw2

radic1minus w2

) (421f)

Kss3 =

w(3minus 2w2)

3radicw

(421g)

3 = K13

ndash 28 ndash

JCAP01(2016)028

-1000 -500 0 500

micro = ai

radick)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

langM2

rang+radic

2GFN`β2max

2+radic

2GFβ2max

intdΓprime

(v minus vprime)2

ω =∆m2

2E= 063 kmminus1

(10 MeV

) (29)

rangrarr ω

2 00 minus1

) (210)

ndash 7 ndash

JCAP01(2016)028

radic2GFN`β

2rarr λv2

2 00 minus1

) (211)

λ =radic

2GFne(r)R2

1012 g cmminus3

r2 (212)

micro =radic

2GFnνe(r)R2

4times1052 ergs

10 MeV

〈Eνe〉

(30 km

)2 (Rr

txωv =Tr(txωv)

2+nνe2

Gtxωv (214)

ndash 8 ndash

JCAP01(2016)028

Htxωv =(ω + λx

12v2)(+1

2 00 minus1

)+ micro

intdΓprime

(v minus vprime)2

Gtxωprimevprime

2 (216)

25 Mass ordering

26 Linearization

(g GGlowast minusg

)(218)

ndash 9 ndash

JCAP01(2016)028

[ω + λx

12v2 + micro

intdΓprime 1

]Gtxωv

minus gtxωv[micro

intdΓprime 1

] (219b)

intdΓ gωv ε1 =

ω + 12λv2 + 1

Stxωv =Gtxωvgωv

intdΓprime 1

ndash 10 ndash

JCAP01(2016)028

iStkωv =(ω + 1

intdωprime

intdvprime 1

intinfin0 dω

intdv gωv = 1 In

)QΩkωv = micro

intdωprime

intdvprime 1

gωv = hω fv (227)

ndash 11 ndash

JCAP01(2016)028

)QΩkωv = micro

minus Ω

= 0 (31)

)micro2 = 1 (32)

ndash 12 ndash

JCAP01(2016)028

[(1 + α)ω0 + (1minus α)Ω

]= 0 (33)

2plusmn

radic[ω0 +

(1minus α)micro

(1 +radicα)2

ltmicro

(1minusradicα)2

κmax =2radicα

1minus αω0 (36)

(1minus α)2 (37)

1minus 2radicα

1 + αlt

microκmax

lt 1 +2radicα

1 + α (38)

radicω0k

minusω0 minus Ω

)m2 = 1 (39)

ndash 13 ndash

JCAP01(2016)028

ω0= minusε

4plusmnradic

4 (310)

micro2κmax

=4 (1 + α)

ndash 14 ndash

JCAP01(2016)028

int +infin

int +1

12 λ v

2 + k v + ω minus Ω(313)

A0 +A1v +A2v2 =

int +infin

int +1

A0 +A1vprime +A2v

prime2

12 λ v

I2 I3 I4

I0 I1 I2

= 0 (315)

In =micro

int +infin

minusinfindω hω

int +1

minus1dv

12 λ v

2 + k v + ω minus Ω (316)

In =micro

2(1 + n)

ω0 + Ω+

ω0 minus Ω

2(1 + n)

ω20 minus Ω2

ndash 15 ndash

JCAP01(2016)028

13 0 1

50 minus2

3 01 0 1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (318)

microrarr micro

30times

radic5 gt 0

minus10 lt 0

5minus 3radic

5 lt 0

176 lt microω0 lt 5974 (320a)

ndash 16 ndash

JCAP01(2016)028

GrowthRateκ

k = 0 λ = 0 nv = 1

GrowthRateκ

k = 0 λ = 0 nv = 2

-500 -400 -300 -200 -100 0 1000

GrowthRateκ

k = 0 λ = 0 nv = 10

ndash 17 ndash

JCAP01(2016)028

-800 -600 -400 -200 00

GrowthRateκ

5101520

-150 -100 -50 0 50 100 1500

GrowthRateκ

1010 2020

-3000 -2500 -2000 -1500 -1000 -500 00

GrowthRateκ

255075100

-400 -200 0 200 4000

GrowthRateκ

2525 5050 7575 100100

ndash 18 ndash

JCAP01(2016)028

I2 0 I4

0 minus2I2 0I0 0 I2

= 0 (321)

ndash 19 ndash

JCAP01(2016)028

GrowthRateκ

k = 25 nv = 2

GrowthRateκ

k = 25 nv = 5

GrowthRateκ

k = 25 nv = 10

-1200-1000 -800 -600 -400 -200 0 2000

GrowthRateκ

k = 25 nv = 20

ndash 20 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

ndash 21 ndash

JCAP01(2016)028

int +infin

int +π

ndash 22 ndash

JCAP01(2016)028

λndash

k = 01041

k = 0 102 103 104

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

= 0 (43)

Ia = micro

int +infin

minusinfindω hω

int +π

minusπdϕ

fa(ϕ)

ndash 23 ndash

JCAP01(2016)028

1 0 00 minus1

2 00 0 minus1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (45)

Ia =micro

int +infin

minusinfindω hωFa

(ω minus Ω

)where Fa(w) =

int +π

minusπdϕ

fa(ϕ)

cosϕ+ w (46)

Ia =micro

(ω0 minus Ω

)minus αFa

(minusω0 minus Ω

)] (47)

radicw + 1 (48)

s(w) Fc = 1minus w

s(w) Fcc = minusw +

ndash 24 ndash

JCAP01(2016)028

-3000 -2000 -1000 0 1000 2000

micro (410)

radicω0

radicω0 minus Ω

ndash 25 ndash

JCAP01(2016)028

intdv = (1π)

int +πminusπ dϕ

= micro

int +infin

int +π

minusπ

dϕprime

0dvprime vprime

QΩkωvϕ =A0 +A2v

2 + k v cϕ + ω minus Ω (412)

3 I15 Ic

1 I13 Ic

2 0minus2Ic

3 00 0 0 minus2Iss

= 0 (413)

integrals are now

Ian = micro

int +infin

minusinfindω hω

int +π

minusπ

vn fa(ϕ)12 λv

2 + k v cϕ + ω minus Ω (414)

3 = I13 minus Icc

= 0 (415)

In = micro

int +infin

12 λv

2 + ω minus Ω (416)

ndash 26 ndash

JCAP01(2016)028

2(ω minus Ω)

) (417a)

2(ω minus Ω)

)](417b)

(2(ω minus Ω)

2(ω minus Ω)

)] (417c)

127 lt microω0 lt 4328 (419a)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v cϕ + w(420)

K11 = w +

radic1minus w

(421a)

radicminuswradic

1minus w2 (1 + 2w2)

3radicw

(421b)

radicminuswradic

1minus w2 (3 + 4w2 + 8w4)

15radicw

(421c)

2minus w2 minus

radicminusw2

ndash 27 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

4minus 2w4 +

radicminusw2radic

1minus w2 (1 + 2w2)

3 (421e)

Kcc3 = minusw

2minus w2 minus

radicminusw2

radic1minus w2

) (421f)

Kss3 =

w(3minus 2w2)

3radicw

(421g)

3 = K13

ndash 28 ndash

JCAP01(2016)028

-1000 -500 0 500

micro = ai

radick)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

radic2GFN`β

2rarr λv2

2 00 minus1

) (211)

λ =radic

2GFne(r)R2

1012 g cmminus3

r2 (212)

micro =radic

2GFnνe(r)R2

4times1052 ergs

10 MeV

〈Eνe〉

(30 km

)2 (Rr

txωv =Tr(txωv)

2+nνe2

Gtxωv (214)

ndash 8 ndash

JCAP01(2016)028

Htxωv =(ω + λx

12v2)(+1

2 00 minus1

)+ micro

intdΓprime

(v minus vprime)2

Gtxωprimevprime

2 (216)

25 Mass ordering

26 Linearization

(g GGlowast minusg

)(218)

ndash 9 ndash

JCAP01(2016)028

[ω + λx

12v2 + micro

intdΓprime 1

]Gtxωv

minus gtxωv[micro

intdΓprime 1

] (219b)

intdΓ gωv ε1 =

ω + 12λv2 + 1

Stxωv =Gtxωvgωv

intdΓprime 1

ndash 10 ndash

JCAP01(2016)028

iStkωv =(ω + 1

intdωprime

intdvprime 1

intinfin0 dω

intdv gωv = 1 In

)QΩkωv = micro

intdωprime

intdvprime 1

gωv = hω fv (227)

ndash 11 ndash

JCAP01(2016)028

)QΩkωv = micro

minus Ω

= 0 (31)

)micro2 = 1 (32)

ndash 12 ndash

JCAP01(2016)028

[(1 + α)ω0 + (1minus α)Ω

]= 0 (33)

2plusmn

radic[ω0 +

(1minus α)micro

(1 +radicα)2

ltmicro

(1minusradicα)2

κmax =2radicα

1minus αω0 (36)

(1minus α)2 (37)

1minus 2radicα

1 + αlt

microκmax

lt 1 +2radicα

1 + α (38)

radicω0k

minusω0 minus Ω

)m2 = 1 (39)

ndash 13 ndash

JCAP01(2016)028

ω0= minusε

4plusmnradic

4 (310)

micro2κmax

=4 (1 + α)

ndash 14 ndash

JCAP01(2016)028

int +infin

int +1

12 λ v

2 + k v + ω minus Ω(313)

A0 +A1v +A2v2 =

int +infin

int +1

A0 +A1vprime +A2v

prime2

12 λ v

I2 I3 I4

I0 I1 I2

= 0 (315)

In =micro

int +infin

minusinfindω hω

int +1

minus1dv

12 λ v

2 + k v + ω minus Ω (316)

In =micro

2(1 + n)

ω0 + Ω+

ω0 minus Ω

2(1 + n)

ω20 minus Ω2

ndash 15 ndash

JCAP01(2016)028

13 0 1

50 minus2

3 01 0 1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (318)

microrarr micro

30times

radic5 gt 0

minus10 lt 0

5minus 3radic

5 lt 0

176 lt microω0 lt 5974 (320a)

ndash 16 ndash

JCAP01(2016)028

GrowthRateκ

k = 0 λ = 0 nv = 1

GrowthRateκ

k = 0 λ = 0 nv = 2

-500 -400 -300 -200 -100 0 1000

GrowthRateκ

k = 0 λ = 0 nv = 10

ndash 17 ndash

JCAP01(2016)028

-800 -600 -400 -200 00

GrowthRateκ

5101520

-150 -100 -50 0 50 100 1500

GrowthRateκ

1010 2020

-3000 -2500 -2000 -1500 -1000 -500 00

GrowthRateκ

255075100

-400 -200 0 200 4000

GrowthRateκ

2525 5050 7575 100100

ndash 18 ndash

JCAP01(2016)028

I2 0 I4

0 minus2I2 0I0 0 I2

= 0 (321)

ndash 19 ndash

JCAP01(2016)028

GrowthRateκ

k = 25 nv = 2

GrowthRateκ

k = 25 nv = 5

GrowthRateκ

k = 25 nv = 10

-1200-1000 -800 -600 -400 -200 0 2000

GrowthRateκ

k = 25 nv = 20

ndash 20 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

ndash 21 ndash

JCAP01(2016)028

int +infin

int +π

ndash 22 ndash

JCAP01(2016)028

λndash

k = 01041

k = 0 102 103 104

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

= 0 (43)

Ia = micro

int +infin

minusinfindω hω

int +π

minusπdϕ

fa(ϕ)

ndash 23 ndash

JCAP01(2016)028

1 0 00 minus1

2 00 0 minus1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (45)

Ia =micro

int +infin

minusinfindω hωFa

(ω minus Ω

)where Fa(w) =

int +π

minusπdϕ

fa(ϕ)

cosϕ+ w (46)

Ia =micro

(ω0 minus Ω

)minus αFa

(minusω0 minus Ω

)] (47)

radicw + 1 (48)

s(w) Fc = 1minus w

s(w) Fcc = minusw +

ndash 24 ndash

JCAP01(2016)028

-3000 -2000 -1000 0 1000 2000

micro (410)

radicω0

radicω0 minus Ω

ndash 25 ndash

JCAP01(2016)028

intdv = (1π)

int +πminusπ dϕ

= micro

int +infin

int +π

minusπ

dϕprime

0dvprime vprime

QΩkωvϕ =A0 +A2v

2 + k v cϕ + ω minus Ω (412)

3 I15 Ic

1 I13 Ic

2 0minus2Ic

3 00 0 0 minus2Iss

= 0 (413)

integrals are now

Ian = micro

int +infin

minusinfindω hω

int +π

minusπ

vn fa(ϕ)12 λv

2 + k v cϕ + ω minus Ω (414)

3 = I13 minus Icc

= 0 (415)

In = micro

int +infin

12 λv

2 + ω minus Ω (416)

ndash 26 ndash

JCAP01(2016)028

2(ω minus Ω)

) (417a)

2(ω minus Ω)

)](417b)

(2(ω minus Ω)

2(ω minus Ω)

)] (417c)

127 lt microω0 lt 4328 (419a)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v cϕ + w(420)

K11 = w +

radic1minus w

(421a)

radicminuswradic

1minus w2 (1 + 2w2)

3radicw

(421b)

radicminuswradic

1minus w2 (3 + 4w2 + 8w4)

15radicw

(421c)

2minus w2 minus

radicminusw2

ndash 27 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

4minus 2w4 +

radicminusw2radic

1minus w2 (1 + 2w2)

3 (421e)

Kcc3 = minusw

2minus w2 minus

radicminusw2

radic1minus w2

) (421f)

Kss3 =

w(3minus 2w2)

3radicw

(421g)

3 = K13

ndash 28 ndash

JCAP01(2016)028

-1000 -500 0 500

micro = ai

radick)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

Htxωv =(ω + λx

12v2)(+1

2 00 minus1

)+ micro

intdΓprime

(v minus vprime)2

Gtxωprimevprime

2 (216)

25 Mass ordering

26 Linearization

(g GGlowast minusg

)(218)

ndash 9 ndash

JCAP01(2016)028

[ω + λx

12v2 + micro

intdΓprime 1

]Gtxωv

minus gtxωv[micro

intdΓprime 1

] (219b)

intdΓ gωv ε1 =

ω + 12λv2 + 1

Stxωv =Gtxωvgωv

intdΓprime 1

ndash 10 ndash

JCAP01(2016)028

iStkωv =(ω + 1

intdωprime

intdvprime 1

intinfin0 dω

intdv gωv = 1 In

)QΩkωv = micro

intdωprime

intdvprime 1

gωv = hω fv (227)

ndash 11 ndash

JCAP01(2016)028

)QΩkωv = micro

minus Ω

= 0 (31)

)micro2 = 1 (32)

ndash 12 ndash

JCAP01(2016)028

[(1 + α)ω0 + (1minus α)Ω

]= 0 (33)

2plusmn

radic[ω0 +

(1minus α)micro

(1 +radicα)2

ltmicro

(1minusradicα)2

κmax =2radicα

1minus αω0 (36)

(1minus α)2 (37)

1minus 2radicα

1 + αlt

microκmax

lt 1 +2radicα

1 + α (38)

radicω0k

minusω0 minus Ω

)m2 = 1 (39)

ndash 13 ndash

JCAP01(2016)028

ω0= minusε

4plusmnradic

4 (310)

micro2κmax

=4 (1 + α)

ndash 14 ndash

JCAP01(2016)028

int +infin

int +1

12 λ v

2 + k v + ω minus Ω(313)

A0 +A1v +A2v2 =

int +infin

int +1

A0 +A1vprime +A2v

prime2

12 λ v

I2 I3 I4

I0 I1 I2

= 0 (315)

In =micro

int +infin

minusinfindω hω

int +1

minus1dv

12 λ v

2 + k v + ω minus Ω (316)

In =micro

2(1 + n)

ω0 + Ω+

ω0 minus Ω

2(1 + n)

ω20 minus Ω2

ndash 15 ndash

JCAP01(2016)028

13 0 1

50 minus2

3 01 0 1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (318)

microrarr micro

30times

radic5 gt 0

minus10 lt 0

5minus 3radic

5 lt 0

176 lt microω0 lt 5974 (320a)

ndash 16 ndash

JCAP01(2016)028

GrowthRateκ

k = 0 λ = 0 nv = 1

GrowthRateκ

k = 0 λ = 0 nv = 2

-500 -400 -300 -200 -100 0 1000

GrowthRateκ

k = 0 λ = 0 nv = 10

ndash 17 ndash

JCAP01(2016)028

-800 -600 -400 -200 00

GrowthRateκ

5101520

-150 -100 -50 0 50 100 1500

GrowthRateκ

1010 2020

-3000 -2500 -2000 -1500 -1000 -500 00

GrowthRateκ

255075100

-400 -200 0 200 4000

GrowthRateκ

2525 5050 7575 100100

ndash 18 ndash

JCAP01(2016)028

I2 0 I4

0 minus2I2 0I0 0 I2

= 0 (321)

ndash 19 ndash

JCAP01(2016)028

GrowthRateκ

k = 25 nv = 2

GrowthRateκ

k = 25 nv = 5

GrowthRateκ

k = 25 nv = 10

-1200-1000 -800 -600 -400 -200 0 2000

GrowthRateκ

k = 25 nv = 20

ndash 20 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

ndash 21 ndash

JCAP01(2016)028

int +infin

int +π

ndash 22 ndash

JCAP01(2016)028

λndash

k = 01041

k = 0 102 103 104

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

= 0 (43)

Ia = micro

int +infin

minusinfindω hω

int +π

minusπdϕ

fa(ϕ)

ndash 23 ndash

JCAP01(2016)028

1 0 00 minus1

2 00 0 minus1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (45)

Ia =micro

int +infin

minusinfindω hωFa

(ω minus Ω

)where Fa(w) =

int +π

minusπdϕ

fa(ϕ)

cosϕ+ w (46)

Ia =micro

(ω0 minus Ω

)minus αFa

(minusω0 minus Ω

)] (47)

radicw + 1 (48)

s(w) Fc = 1minus w

s(w) Fcc = minusw +

ndash 24 ndash

JCAP01(2016)028

-3000 -2000 -1000 0 1000 2000

micro (410)

radicω0

radicω0 minus Ω

ndash 25 ndash

JCAP01(2016)028

intdv = (1π)

int +πminusπ dϕ

= micro

int +infin

int +π

minusπ

dϕprime

0dvprime vprime

QΩkωvϕ =A0 +A2v

2 + k v cϕ + ω minus Ω (412)

3 I15 Ic

1 I13 Ic

2 0minus2Ic

3 00 0 0 minus2Iss

= 0 (413)

integrals are now

Ian = micro

int +infin

minusinfindω hω

int +π

minusπ

vn fa(ϕ)12 λv

2 + k v cϕ + ω minus Ω (414)

3 = I13 minus Icc

= 0 (415)

In = micro

int +infin

12 λv

2 + ω minus Ω (416)

ndash 26 ndash

JCAP01(2016)028

2(ω minus Ω)

) (417a)

2(ω minus Ω)

)](417b)

(2(ω minus Ω)

2(ω minus Ω)

)] (417c)

127 lt microω0 lt 4328 (419a)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v cϕ + w(420)

K11 = w +

radic1minus w

(421a)

radicminuswradic

1minus w2 (1 + 2w2)

3radicw

(421b)

radicminuswradic

1minus w2 (3 + 4w2 + 8w4)

15radicw

(421c)

2minus w2 minus

radicminusw2

ndash 27 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

4minus 2w4 +

radicminusw2radic

1minus w2 (1 + 2w2)

3 (421e)

Kcc3 = minusw

2minus w2 minus

radicminusw2

radic1minus w2

) (421f)

Kss3 =

w(3minus 2w2)

3radicw

(421g)

3 = K13

ndash 28 ndash

JCAP01(2016)028

-1000 -500 0 500

micro = ai

radick)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

[ω + λx

12v2 + micro

intdΓprime 1

]Gtxωv

minus gtxωv[micro

intdΓprime 1

] (219b)

intdΓ gωv ε1 =

ω + 12λv2 + 1

Stxωv =Gtxωvgωv

intdΓprime 1

ndash 10 ndash

JCAP01(2016)028

iStkωv =(ω + 1

intdωprime

intdvprime 1

intinfin0 dω

intdv gωv = 1 In

)QΩkωv = micro

intdωprime

intdvprime 1

gωv = hω fv (227)

ndash 11 ndash

JCAP01(2016)028

)QΩkωv = micro

minus Ω

= 0 (31)

)micro2 = 1 (32)

ndash 12 ndash

JCAP01(2016)028

[(1 + α)ω0 + (1minus α)Ω

]= 0 (33)

2plusmn

radic[ω0 +

(1minus α)micro

(1 +radicα)2

ltmicro

(1minusradicα)2

κmax =2radicα

1minus αω0 (36)

(1minus α)2 (37)

1minus 2radicα

1 + αlt

microκmax

lt 1 +2radicα

1 + α (38)

radicω0k

minusω0 minus Ω

)m2 = 1 (39)

ndash 13 ndash

JCAP01(2016)028

ω0= minusε

4plusmnradic

4 (310)

micro2κmax

=4 (1 + α)

ndash 14 ndash

JCAP01(2016)028

int +infin

int +1

12 λ v

2 + k v + ω minus Ω(313)

A0 +A1v +A2v2 =

int +infin

int +1

A0 +A1vprime +A2v

prime2

12 λ v

I2 I3 I4

I0 I1 I2

= 0 (315)

In =micro

int +infin

minusinfindω hω

int +1

minus1dv

12 λ v

2 + k v + ω minus Ω (316)

In =micro

2(1 + n)

ω0 + Ω+

ω0 minus Ω

2(1 + n)

ω20 minus Ω2

ndash 15 ndash

JCAP01(2016)028

13 0 1

50 minus2

3 01 0 1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (318)

microrarr micro

30times

radic5 gt 0

minus10 lt 0

5minus 3radic

5 lt 0

176 lt microω0 lt 5974 (320a)

ndash 16 ndash

JCAP01(2016)028

GrowthRateκ

k = 0 λ = 0 nv = 1

GrowthRateκ

k = 0 λ = 0 nv = 2

-500 -400 -300 -200 -100 0 1000

GrowthRateκ

k = 0 λ = 0 nv = 10

ndash 17 ndash

JCAP01(2016)028

-800 -600 -400 -200 00

GrowthRateκ

5101520

-150 -100 -50 0 50 100 1500

GrowthRateκ

1010 2020

-3000 -2500 -2000 -1500 -1000 -500 00

GrowthRateκ

255075100

-400 -200 0 200 4000

GrowthRateκ

2525 5050 7575 100100

ndash 18 ndash

JCAP01(2016)028

I2 0 I4

0 minus2I2 0I0 0 I2

= 0 (321)

ndash 19 ndash

JCAP01(2016)028

GrowthRateκ

k = 25 nv = 2

GrowthRateκ

k = 25 nv = 5

GrowthRateκ

k = 25 nv = 10

-1200-1000 -800 -600 -400 -200 0 2000

GrowthRateκ

k = 25 nv = 20

ndash 20 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

ndash 21 ndash

JCAP01(2016)028

int +infin

int +π

ndash 22 ndash

JCAP01(2016)028

λndash

k = 01041

k = 0 102 103 104

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

= 0 (43)

Ia = micro

int +infin

minusinfindω hω

int +π

minusπdϕ

fa(ϕ)

ndash 23 ndash

JCAP01(2016)028

1 0 00 minus1

2 00 0 minus1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (45)

Ia =micro

int +infin

minusinfindω hωFa

(ω minus Ω

)where Fa(w) =

int +π

minusπdϕ

fa(ϕ)

cosϕ+ w (46)

Ia =micro

(ω0 minus Ω

)minus αFa

(minusω0 minus Ω

)] (47)

radicw + 1 (48)

s(w) Fc = 1minus w

s(w) Fcc = minusw +

ndash 24 ndash

JCAP01(2016)028

-3000 -2000 -1000 0 1000 2000

micro (410)

radicω0

radicω0 minus Ω

ndash 25 ndash

JCAP01(2016)028

intdv = (1π)

int +πminusπ dϕ

= micro

int +infin

int +π

minusπ

dϕprime

0dvprime vprime

QΩkωvϕ =A0 +A2v

2 + k v cϕ + ω minus Ω (412)

3 I15 Ic

1 I13 Ic

2 0minus2Ic

3 00 0 0 minus2Iss

= 0 (413)

integrals are now

Ian = micro

int +infin

minusinfindω hω

int +π

minusπ

vn fa(ϕ)12 λv

2 + k v cϕ + ω minus Ω (414)

3 = I13 minus Icc

= 0 (415)

In = micro

int +infin

12 λv

2 + ω minus Ω (416)

ndash 26 ndash

JCAP01(2016)028

2(ω minus Ω)

) (417a)

2(ω minus Ω)

)](417b)

(2(ω minus Ω)

2(ω minus Ω)

)] (417c)

127 lt microω0 lt 4328 (419a)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v cϕ + w(420)

K11 = w +

radic1minus w

(421a)

radicminuswradic

1minus w2 (1 + 2w2)

3radicw

(421b)

radicminuswradic

1minus w2 (3 + 4w2 + 8w4)

15radicw

(421c)

2minus w2 minus

radicminusw2

ndash 27 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

4minus 2w4 +

radicminusw2radic

1minus w2 (1 + 2w2)

3 (421e)

Kcc3 = minusw

2minus w2 minus

radicminusw2

radic1minus w2

) (421f)

Kss3 =

w(3minus 2w2)

3radicw

(421g)

3 = K13

ndash 28 ndash

JCAP01(2016)028

-1000 -500 0 500

micro = ai

radick)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

iStkωv =(ω + 1

intdωprime

intdvprime 1

intinfin0 dω

intdv gωv = 1 In

)QΩkωv = micro

intdωprime

intdvprime 1

gωv = hω fv (227)

ndash 11 ndash

JCAP01(2016)028

)QΩkωv = micro

minus Ω

= 0 (31)

)micro2 = 1 (32)

ndash 12 ndash

JCAP01(2016)028

[(1 + α)ω0 + (1minus α)Ω

]= 0 (33)

2plusmn

radic[ω0 +

(1minus α)micro

(1 +radicα)2

ltmicro

(1minusradicα)2

κmax =2radicα

1minus αω0 (36)

(1minus α)2 (37)

1minus 2radicα

1 + αlt

microκmax

lt 1 +2radicα

1 + α (38)

radicω0k

minusω0 minus Ω

)m2 = 1 (39)

ndash 13 ndash

JCAP01(2016)028

ω0= minusε

4plusmnradic

4 (310)

micro2κmax

=4 (1 + α)

ndash 14 ndash

JCAP01(2016)028

int +infin

int +1

12 λ v

2 + k v + ω minus Ω(313)

A0 +A1v +A2v2 =

int +infin

int +1

A0 +A1vprime +A2v

prime2

12 λ v

I2 I3 I4

I0 I1 I2

= 0 (315)

In =micro

int +infin

minusinfindω hω

int +1

minus1dv

12 λ v

2 + k v + ω minus Ω (316)

In =micro

2(1 + n)

ω0 + Ω+

ω0 minus Ω

2(1 + n)

ω20 minus Ω2

ndash 15 ndash

JCAP01(2016)028

13 0 1

50 minus2

3 01 0 1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (318)

microrarr micro

30times

radic5 gt 0

minus10 lt 0

5minus 3radic

5 lt 0

176 lt microω0 lt 5974 (320a)

ndash 16 ndash

JCAP01(2016)028

GrowthRateκ

k = 0 λ = 0 nv = 1

GrowthRateκ

k = 0 λ = 0 nv = 2

-500 -400 -300 -200 -100 0 1000

GrowthRateκ

k = 0 λ = 0 nv = 10

ndash 17 ndash

JCAP01(2016)028

-800 -600 -400 -200 00

GrowthRateκ

5101520

-150 -100 -50 0 50 100 1500

GrowthRateκ

1010 2020

-3000 -2500 -2000 -1500 -1000 -500 00

GrowthRateκ

255075100

-400 -200 0 200 4000

GrowthRateκ

2525 5050 7575 100100

ndash 18 ndash

JCAP01(2016)028

I2 0 I4

0 minus2I2 0I0 0 I2

= 0 (321)

ndash 19 ndash

JCAP01(2016)028

GrowthRateκ

k = 25 nv = 2

GrowthRateκ

k = 25 nv = 5

GrowthRateκ

k = 25 nv = 10

-1200-1000 -800 -600 -400 -200 0 2000

GrowthRateκ

k = 25 nv = 20

ndash 20 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

ndash 21 ndash

JCAP01(2016)028

int +infin

int +π

ndash 22 ndash

JCAP01(2016)028

λndash

k = 01041

k = 0 102 103 104

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

= 0 (43)

Ia = micro

int +infin

minusinfindω hω

int +π

minusπdϕ

fa(ϕ)

ndash 23 ndash

JCAP01(2016)028

1 0 00 minus1

2 00 0 minus1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (45)

Ia =micro

int +infin

minusinfindω hωFa

(ω minus Ω

)where Fa(w) =

int +π

minusπdϕ

fa(ϕ)

cosϕ+ w (46)

Ia =micro

(ω0 minus Ω

)minus αFa

(minusω0 minus Ω

)] (47)

radicw + 1 (48)

s(w) Fc = 1minus w

s(w) Fcc = minusw +

ndash 24 ndash

JCAP01(2016)028

-3000 -2000 -1000 0 1000 2000

micro (410)

radicω0

radicω0 minus Ω

ndash 25 ndash

JCAP01(2016)028

intdv = (1π)

int +πminusπ dϕ

= micro

int +infin

int +π

minusπ

dϕprime

0dvprime vprime

QΩkωvϕ =A0 +A2v

2 + k v cϕ + ω minus Ω (412)

3 I15 Ic

1 I13 Ic

2 0minus2Ic

3 00 0 0 minus2Iss

= 0 (413)

integrals are now

Ian = micro

int +infin

minusinfindω hω

int +π

minusπ

vn fa(ϕ)12 λv

2 + k v cϕ + ω minus Ω (414)

3 = I13 minus Icc

= 0 (415)

In = micro

int +infin

12 λv

2 + ω minus Ω (416)

ndash 26 ndash

JCAP01(2016)028

2(ω minus Ω)

) (417a)

2(ω minus Ω)

)](417b)

(2(ω minus Ω)

2(ω minus Ω)

)] (417c)

127 lt microω0 lt 4328 (419a)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v cϕ + w(420)

K11 = w +

radic1minus w

(421a)

radicminuswradic

1minus w2 (1 + 2w2)

3radicw

(421b)

radicminuswradic

1minus w2 (3 + 4w2 + 8w4)

15radicw

(421c)

2minus w2 minus

radicminusw2

ndash 27 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

4minus 2w4 +

radicminusw2radic

1minus w2 (1 + 2w2)

3 (421e)

Kcc3 = minusw

2minus w2 minus

radicminusw2

radic1minus w2

) (421f)

Kss3 =

w(3minus 2w2)

3radicw

(421g)

3 = K13

ndash 28 ndash

JCAP01(2016)028

-1000 -500 0 500

micro = ai

radick)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

)QΩkωv = micro

minus Ω

= 0 (31)

)micro2 = 1 (32)

ndash 12 ndash

JCAP01(2016)028

[(1 + α)ω0 + (1minus α)Ω

]= 0 (33)

2plusmn

radic[ω0 +

(1minus α)micro

(1 +radicα)2

ltmicro

(1minusradicα)2

κmax =2radicα

1minus αω0 (36)

(1minus α)2 (37)

1minus 2radicα

1 + αlt

microκmax

lt 1 +2radicα

1 + α (38)

radicω0k

minusω0 minus Ω

)m2 = 1 (39)

ndash 13 ndash

JCAP01(2016)028

ω0= minusε

4plusmnradic

4 (310)

micro2κmax

=4 (1 + α)

ndash 14 ndash

JCAP01(2016)028

int +infin

int +1

12 λ v

2 + k v + ω minus Ω(313)

A0 +A1v +A2v2 =

int +infin

int +1

A0 +A1vprime +A2v

prime2

12 λ v

I2 I3 I4

I0 I1 I2

= 0 (315)

In =micro

int +infin

minusinfindω hω

int +1

minus1dv

12 λ v

2 + k v + ω minus Ω (316)

In =micro

2(1 + n)

ω0 + Ω+

ω0 minus Ω

2(1 + n)

ω20 minus Ω2

ndash 15 ndash

JCAP01(2016)028

13 0 1

50 minus2

3 01 0 1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (318)

microrarr micro

30times

radic5 gt 0

minus10 lt 0

5minus 3radic

5 lt 0

176 lt microω0 lt 5974 (320a)

ndash 16 ndash

JCAP01(2016)028

GrowthRateκ

k = 0 λ = 0 nv = 1

GrowthRateκ

k = 0 λ = 0 nv = 2

-500 -400 -300 -200 -100 0 1000

GrowthRateκ

k = 0 λ = 0 nv = 10

ndash 17 ndash

JCAP01(2016)028

-800 -600 -400 -200 00

GrowthRateκ

5101520

-150 -100 -50 0 50 100 1500

GrowthRateκ

1010 2020

-3000 -2500 -2000 -1500 -1000 -500 00

GrowthRateκ

255075100

-400 -200 0 200 4000

GrowthRateκ

2525 5050 7575 100100

ndash 18 ndash

JCAP01(2016)028

I2 0 I4

0 minus2I2 0I0 0 I2

= 0 (321)

ndash 19 ndash

JCAP01(2016)028

GrowthRateκ

k = 25 nv = 2

GrowthRateκ

k = 25 nv = 5

GrowthRateκ

k = 25 nv = 10

-1200-1000 -800 -600 -400 -200 0 2000

GrowthRateκ

k = 25 nv = 20

ndash 20 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

ndash 21 ndash

JCAP01(2016)028

int +infin

int +π

ndash 22 ndash

JCAP01(2016)028

λndash

k = 01041

k = 0 102 103 104

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

= 0 (43)

Ia = micro

int +infin

minusinfindω hω

int +π

minusπdϕ

fa(ϕ)

ndash 23 ndash

JCAP01(2016)028

1 0 00 minus1

2 00 0 minus1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (45)

Ia =micro

int +infin

minusinfindω hωFa

(ω minus Ω

)where Fa(w) =

int +π

minusπdϕ

fa(ϕ)

cosϕ+ w (46)

Ia =micro

(ω0 minus Ω

)minus αFa

(minusω0 minus Ω

)] (47)

radicw + 1 (48)

s(w) Fc = 1minus w

s(w) Fcc = minusw +

ndash 24 ndash

JCAP01(2016)028

-3000 -2000 -1000 0 1000 2000

micro (410)

radicω0

radicω0 minus Ω

ndash 25 ndash

JCAP01(2016)028

intdv = (1π)

int +πminusπ dϕ

= micro

int +infin

int +π

minusπ

dϕprime

0dvprime vprime

QΩkωvϕ =A0 +A2v

2 + k v cϕ + ω minus Ω (412)

3 I15 Ic

1 I13 Ic

2 0minus2Ic

3 00 0 0 minus2Iss

= 0 (413)

integrals are now

Ian = micro

int +infin

minusinfindω hω

int +π

minusπ

vn fa(ϕ)12 λv

2 + k v cϕ + ω minus Ω (414)

3 = I13 minus Icc

= 0 (415)

In = micro

int +infin

12 λv

2 + ω minus Ω (416)

ndash 26 ndash

JCAP01(2016)028

2(ω minus Ω)

) (417a)

2(ω minus Ω)

)](417b)

(2(ω minus Ω)

2(ω minus Ω)

)] (417c)

127 lt microω0 lt 4328 (419a)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v cϕ + w(420)

K11 = w +

radic1minus w

(421a)

radicminuswradic

1minus w2 (1 + 2w2)

3radicw

(421b)

radicminuswradic

1minus w2 (3 + 4w2 + 8w4)

15radicw

(421c)

2minus w2 minus

radicminusw2

ndash 27 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

4minus 2w4 +

radicminusw2radic

1minus w2 (1 + 2w2)

3 (421e)

Kcc3 = minusw

2minus w2 minus

radicminusw2

radic1minus w2

) (421f)

Kss3 =

w(3minus 2w2)

3radicw

(421g)

3 = K13

ndash 28 ndash

JCAP01(2016)028

-1000 -500 0 500

micro = ai

radick)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

[(1 + α)ω0 + (1minus α)Ω

]= 0 (33)

2plusmn

radic[ω0 +

(1minus α)micro

(1 +radicα)2

ltmicro

(1minusradicα)2

κmax =2radicα

1minus αω0 (36)

(1minus α)2 (37)

1minus 2radicα

1 + αlt

microκmax

lt 1 +2radicα

1 + α (38)

radicω0k

minusω0 minus Ω

)m2 = 1 (39)

ndash 13 ndash

JCAP01(2016)028

ω0= minusε

4plusmnradic

4 (310)

micro2κmax

=4 (1 + α)

ndash 14 ndash

JCAP01(2016)028

int +infin

int +1

12 λ v

2 + k v + ω minus Ω(313)

A0 +A1v +A2v2 =

int +infin

int +1

A0 +A1vprime +A2v

prime2

12 λ v

I2 I3 I4

I0 I1 I2

= 0 (315)

In =micro

int +infin

minusinfindω hω

int +1

minus1dv

12 λ v

2 + k v + ω minus Ω (316)

In =micro

2(1 + n)

ω0 + Ω+

ω0 minus Ω

2(1 + n)

ω20 minus Ω2

ndash 15 ndash

JCAP01(2016)028

13 0 1

50 minus2

3 01 0 1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (318)

microrarr micro

30times

radic5 gt 0

minus10 lt 0

5minus 3radic

5 lt 0

176 lt microω0 lt 5974 (320a)

ndash 16 ndash

JCAP01(2016)028

GrowthRateκ

k = 0 λ = 0 nv = 1

GrowthRateκ

k = 0 λ = 0 nv = 2

-500 -400 -300 -200 -100 0 1000

GrowthRateκ

k = 0 λ = 0 nv = 10

ndash 17 ndash

JCAP01(2016)028

-800 -600 -400 -200 00

GrowthRateκ

5101520

-150 -100 -50 0 50 100 1500

GrowthRateκ

1010 2020

-3000 -2500 -2000 -1500 -1000 -500 00

GrowthRateκ

255075100

-400 -200 0 200 4000

GrowthRateκ

2525 5050 7575 100100

ndash 18 ndash

JCAP01(2016)028

I2 0 I4

0 minus2I2 0I0 0 I2

= 0 (321)

ndash 19 ndash

JCAP01(2016)028

GrowthRateκ

k = 25 nv = 2

GrowthRateκ

k = 25 nv = 5

GrowthRateκ

k = 25 nv = 10

-1200-1000 -800 -600 -400 -200 0 2000

GrowthRateκ

k = 25 nv = 20

ndash 20 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

ndash 21 ndash

JCAP01(2016)028

int +infin

int +π

ndash 22 ndash

JCAP01(2016)028

λndash

k = 01041

k = 0 102 103 104

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

= 0 (43)

Ia = micro

int +infin

minusinfindω hω

int +π

minusπdϕ

fa(ϕ)

ndash 23 ndash

JCAP01(2016)028

1 0 00 minus1

2 00 0 minus1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (45)

Ia =micro

int +infin

minusinfindω hωFa

(ω minus Ω

)where Fa(w) =

int +π

minusπdϕ

fa(ϕ)

cosϕ+ w (46)

Ia =micro

(ω0 minus Ω

)minus αFa

(minusω0 minus Ω

)] (47)

radicw + 1 (48)

s(w) Fc = 1minus w

s(w) Fcc = minusw +

ndash 24 ndash

JCAP01(2016)028

-3000 -2000 -1000 0 1000 2000

micro (410)

radicω0

radicω0 minus Ω

ndash 25 ndash

JCAP01(2016)028

intdv = (1π)

int +πminusπ dϕ

= micro

int +infin

int +π

minusπ

dϕprime

0dvprime vprime

QΩkωvϕ =A0 +A2v

2 + k v cϕ + ω minus Ω (412)

3 I15 Ic

1 I13 Ic

2 0minus2Ic

3 00 0 0 minus2Iss

= 0 (413)

integrals are now

Ian = micro

int +infin

minusinfindω hω

int +π

minusπ

vn fa(ϕ)12 λv

2 + k v cϕ + ω minus Ω (414)

3 = I13 minus Icc

= 0 (415)

In = micro

int +infin

12 λv

2 + ω minus Ω (416)

ndash 26 ndash

JCAP01(2016)028

2(ω minus Ω)

) (417a)

2(ω minus Ω)

)](417b)

(2(ω minus Ω)

2(ω minus Ω)

)] (417c)

127 lt microω0 lt 4328 (419a)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v cϕ + w(420)

K11 = w +

radic1minus w

(421a)

radicminuswradic

1minus w2 (1 + 2w2)

3radicw

(421b)

radicminuswradic

1minus w2 (3 + 4w2 + 8w4)

15radicw

(421c)

2minus w2 minus

radicminusw2

ndash 27 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

4minus 2w4 +

radicminusw2radic

1minus w2 (1 + 2w2)

3 (421e)

Kcc3 = minusw

2minus w2 minus

radicminusw2

radic1minus w2

) (421f)

Kss3 =

w(3minus 2w2)

3radicw

(421g)

3 = K13

ndash 28 ndash

JCAP01(2016)028

-1000 -500 0 500

micro = ai

radick)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

ω0= minusε

4plusmnradic

4 (310)

micro2κmax

=4 (1 + α)

ndash 14 ndash

JCAP01(2016)028

int +infin

int +1

12 λ v

2 + k v + ω minus Ω(313)

A0 +A1v +A2v2 =

int +infin

int +1

A0 +A1vprime +A2v

prime2

12 λ v

I2 I3 I4

I0 I1 I2

= 0 (315)

In =micro

int +infin

minusinfindω hω

int +1

minus1dv

12 λ v

2 + k v + ω minus Ω (316)

In =micro

2(1 + n)

ω0 + Ω+

ω0 minus Ω

2(1 + n)

ω20 minus Ω2

ndash 15 ndash

JCAP01(2016)028

13 0 1

50 minus2

3 01 0 1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (318)

microrarr micro

30times

radic5 gt 0

minus10 lt 0

5minus 3radic

5 lt 0

176 lt microω0 lt 5974 (320a)

ndash 16 ndash

JCAP01(2016)028

GrowthRateκ

k = 0 λ = 0 nv = 1

GrowthRateκ

k = 0 λ = 0 nv = 2

-500 -400 -300 -200 -100 0 1000

GrowthRateκ

k = 0 λ = 0 nv = 10

ndash 17 ndash

JCAP01(2016)028

-800 -600 -400 -200 00

GrowthRateκ

5101520

-150 -100 -50 0 50 100 1500

GrowthRateκ

1010 2020

-3000 -2500 -2000 -1500 -1000 -500 00

GrowthRateκ

255075100

-400 -200 0 200 4000

GrowthRateκ

2525 5050 7575 100100

ndash 18 ndash

JCAP01(2016)028

I2 0 I4

0 minus2I2 0I0 0 I2

= 0 (321)

ndash 19 ndash

JCAP01(2016)028

GrowthRateκ

k = 25 nv = 2

GrowthRateκ

k = 25 nv = 5

GrowthRateκ

k = 25 nv = 10

-1200-1000 -800 -600 -400 -200 0 2000

GrowthRateκ

k = 25 nv = 20

ndash 20 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

ndash 21 ndash

JCAP01(2016)028

int +infin

int +π

ndash 22 ndash

JCAP01(2016)028

λndash

k = 01041

k = 0 102 103 104

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

= 0 (43)

Ia = micro

int +infin

minusinfindω hω

int +π

minusπdϕ

fa(ϕ)

ndash 23 ndash

JCAP01(2016)028

1 0 00 minus1

2 00 0 minus1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (45)

Ia =micro

int +infin

minusinfindω hωFa

(ω minus Ω

)where Fa(w) =

int +π

minusπdϕ

fa(ϕ)

cosϕ+ w (46)

Ia =micro

(ω0 minus Ω

)minus αFa

(minusω0 minus Ω

)] (47)

radicw + 1 (48)

s(w) Fc = 1minus w

s(w) Fcc = minusw +

ndash 24 ndash

JCAP01(2016)028

-3000 -2000 -1000 0 1000 2000

micro (410)

radicω0

radicω0 minus Ω

ndash 25 ndash

JCAP01(2016)028

intdv = (1π)

int +πminusπ dϕ

= micro

int +infin

int +π

minusπ

dϕprime

0dvprime vprime

QΩkωvϕ =A0 +A2v

2 + k v cϕ + ω minus Ω (412)

3 I15 Ic

1 I13 Ic

2 0minus2Ic

3 00 0 0 minus2Iss

= 0 (413)

integrals are now

Ian = micro

int +infin

minusinfindω hω

int +π

minusπ

vn fa(ϕ)12 λv

2 + k v cϕ + ω minus Ω (414)

3 = I13 minus Icc

= 0 (415)

In = micro

int +infin

12 λv

2 + ω minus Ω (416)

ndash 26 ndash

JCAP01(2016)028

2(ω minus Ω)

) (417a)

2(ω minus Ω)

)](417b)

(2(ω minus Ω)

2(ω minus Ω)

)] (417c)

127 lt microω0 lt 4328 (419a)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v cϕ + w(420)

K11 = w +

radic1minus w

(421a)

radicminuswradic

1minus w2 (1 + 2w2)

3radicw

(421b)

radicminuswradic

1minus w2 (3 + 4w2 + 8w4)

15radicw

(421c)

2minus w2 minus

radicminusw2

ndash 27 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

4minus 2w4 +

radicminusw2radic

1minus w2 (1 + 2w2)

3 (421e)

Kcc3 = minusw

2minus w2 minus

radicminusw2

radic1minus w2

) (421f)

Kss3 =

w(3minus 2w2)

3radicw

(421g)

3 = K13

ndash 28 ndash

JCAP01(2016)028

-1000 -500 0 500

micro = ai

radick)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

int +infin

int +1

12 λ v

2 + k v + ω minus Ω(313)

A0 +A1v +A2v2 =

int +infin

int +1

A0 +A1vprime +A2v

prime2

12 λ v

I2 I3 I4

I0 I1 I2

= 0 (315)

In =micro

int +infin

minusinfindω hω

int +1

minus1dv

12 λ v

2 + k v + ω minus Ω (316)

In =micro

2(1 + n)

ω0 + Ω+

ω0 minus Ω

2(1 + n)

ω20 minus Ω2

ndash 15 ndash

JCAP01(2016)028

13 0 1

50 minus2

3 01 0 1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (318)

microrarr micro

30times

radic5 gt 0

minus10 lt 0

5minus 3radic

5 lt 0

176 lt microω0 lt 5974 (320a)

ndash 16 ndash

JCAP01(2016)028

GrowthRateκ

k = 0 λ = 0 nv = 1

GrowthRateκ

k = 0 λ = 0 nv = 2

-500 -400 -300 -200 -100 0 1000

GrowthRateκ

k = 0 λ = 0 nv = 10

ndash 17 ndash

JCAP01(2016)028

-800 -600 -400 -200 00

GrowthRateκ

5101520

-150 -100 -50 0 50 100 1500

GrowthRateκ

1010 2020

-3000 -2500 -2000 -1500 -1000 -500 00

GrowthRateκ

255075100

-400 -200 0 200 4000

GrowthRateκ

2525 5050 7575 100100

ndash 18 ndash

JCAP01(2016)028

I2 0 I4

0 minus2I2 0I0 0 I2

= 0 (321)

ndash 19 ndash

JCAP01(2016)028

GrowthRateκ

k = 25 nv = 2

GrowthRateκ

k = 25 nv = 5

GrowthRateκ

k = 25 nv = 10

-1200-1000 -800 -600 -400 -200 0 2000

GrowthRateκ

k = 25 nv = 20

ndash 20 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

ndash 21 ndash

JCAP01(2016)028

int +infin

int +π

ndash 22 ndash

JCAP01(2016)028

λndash

k = 01041

k = 0 102 103 104

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

= 0 (43)

Ia = micro

int +infin

minusinfindω hω

int +π

minusπdϕ

fa(ϕ)

ndash 23 ndash

JCAP01(2016)028

1 0 00 minus1

2 00 0 minus1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (45)

Ia =micro

int +infin

minusinfindω hωFa

(ω minus Ω

)where Fa(w) =

int +π

minusπdϕ

fa(ϕ)

cosϕ+ w (46)

Ia =micro

(ω0 minus Ω

)minus αFa

(minusω0 minus Ω

)] (47)

radicw + 1 (48)

s(w) Fc = 1minus w

s(w) Fcc = minusw +

ndash 24 ndash

JCAP01(2016)028

-3000 -2000 -1000 0 1000 2000

micro (410)

radicω0

radicω0 minus Ω

ndash 25 ndash

JCAP01(2016)028

intdv = (1π)

int +πminusπ dϕ

= micro

int +infin

int +π

minusπ

dϕprime

0dvprime vprime

QΩkωvϕ =A0 +A2v

2 + k v cϕ + ω minus Ω (412)

3 I15 Ic

1 I13 Ic

2 0minus2Ic

3 00 0 0 minus2Iss

= 0 (413)

integrals are now

Ian = micro

int +infin

minusinfindω hω

int +π

minusπ

vn fa(ϕ)12 λv

2 + k v cϕ + ω minus Ω (414)

3 = I13 minus Icc

= 0 (415)

In = micro

int +infin

12 λv

2 + ω minus Ω (416)

ndash 26 ndash

JCAP01(2016)028

2(ω minus Ω)

) (417a)

2(ω minus Ω)

)](417b)

(2(ω minus Ω)

2(ω minus Ω)

)] (417c)

127 lt microω0 lt 4328 (419a)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v cϕ + w(420)

K11 = w +

radic1minus w

(421a)

radicminuswradic

1minus w2 (1 + 2w2)

3radicw

(421b)

radicminuswradic

1minus w2 (3 + 4w2 + 8w4)

15radicw

(421c)

2minus w2 minus

radicminusw2

ndash 27 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

4minus 2w4 +

radicminusw2radic

1minus w2 (1 + 2w2)

3 (421e)

Kcc3 = minusw

2minus w2 minus

radicminusw2

radic1minus w2

) (421f)

Kss3 =

w(3minus 2w2)

3radicw

(421g)

3 = K13

ndash 28 ndash

JCAP01(2016)028

-1000 -500 0 500

micro = ai

radick)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

13 0 1

50 minus2

3 01 0 1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (318)

microrarr micro

30times

radic5 gt 0

minus10 lt 0

5minus 3radic

5 lt 0

176 lt microω0 lt 5974 (320a)

ndash 16 ndash

JCAP01(2016)028

GrowthRateκ

k = 0 λ = 0 nv = 1

GrowthRateκ

k = 0 λ = 0 nv = 2

-500 -400 -300 -200 -100 0 1000

GrowthRateκ

k = 0 λ = 0 nv = 10

ndash 17 ndash

JCAP01(2016)028

-800 -600 -400 -200 00

GrowthRateκ

5101520

-150 -100 -50 0 50 100 1500

GrowthRateκ

1010 2020

-3000 -2500 -2000 -1500 -1000 -500 00

GrowthRateκ

255075100

-400 -200 0 200 4000

GrowthRateκ

2525 5050 7575 100100

ndash 18 ndash

JCAP01(2016)028

I2 0 I4

0 minus2I2 0I0 0 I2

= 0 (321)

ndash 19 ndash

JCAP01(2016)028

GrowthRateκ

k = 25 nv = 2

GrowthRateκ

k = 25 nv = 5

GrowthRateκ

k = 25 nv = 10

-1200-1000 -800 -600 -400 -200 0 2000

GrowthRateκ

k = 25 nv = 20

ndash 20 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

ndash 21 ndash

JCAP01(2016)028

int +infin

int +π

ndash 22 ndash

JCAP01(2016)028

λndash

k = 01041

k = 0 102 103 104

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

= 0 (43)

Ia = micro

int +infin

minusinfindω hω

int +π

minusπdϕ

fa(ϕ)

ndash 23 ndash

JCAP01(2016)028

1 0 00 minus1

2 00 0 minus1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (45)

Ia =micro

int +infin

minusinfindω hωFa

(ω minus Ω

)where Fa(w) =

int +π

minusπdϕ

fa(ϕ)

cosϕ+ w (46)

Ia =micro

(ω0 minus Ω

)minus αFa

(minusω0 minus Ω

)] (47)

radicw + 1 (48)

s(w) Fc = 1minus w

s(w) Fcc = minusw +

ndash 24 ndash

JCAP01(2016)028

-3000 -2000 -1000 0 1000 2000

micro (410)

radicω0

radicω0 minus Ω

ndash 25 ndash

JCAP01(2016)028

intdv = (1π)

int +πminusπ dϕ

= micro

int +infin

int +π

minusπ

dϕprime

0dvprime vprime

QΩkωvϕ =A0 +A2v

2 + k v cϕ + ω minus Ω (412)

3 I15 Ic

1 I13 Ic

2 0minus2Ic

3 00 0 0 minus2Iss

= 0 (413)

integrals are now

Ian = micro

int +infin

minusinfindω hω

int +π

minusπ

vn fa(ϕ)12 λv

2 + k v cϕ + ω minus Ω (414)

3 = I13 minus Icc

= 0 (415)

In = micro

int +infin

12 λv

2 + ω minus Ω (416)

ndash 26 ndash

JCAP01(2016)028

2(ω minus Ω)

) (417a)

2(ω minus Ω)

)](417b)

(2(ω minus Ω)

2(ω minus Ω)

)] (417c)

127 lt microω0 lt 4328 (419a)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v cϕ + w(420)

K11 = w +

radic1minus w

(421a)

radicminuswradic

1minus w2 (1 + 2w2)

3radicw

(421b)

radicminuswradic

1minus w2 (3 + 4w2 + 8w4)

15radicw

(421c)

2minus w2 minus

radicminusw2

ndash 27 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

4minus 2w4 +

radicminusw2radic

1minus w2 (1 + 2w2)

3 (421e)

Kcc3 = minusw

2minus w2 minus

radicminusw2

radic1minus w2

) (421f)

Kss3 =

w(3minus 2w2)

3radicw

(421g)

3 = K13

ndash 28 ndash

JCAP01(2016)028

-1000 -500 0 500

micro = ai

radick)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

GrowthRateκ

k = 0 λ = 0 nv = 1

GrowthRateκ

k = 0 λ = 0 nv = 2

-500 -400 -300 -200 -100 0 1000

GrowthRateκ

k = 0 λ = 0 nv = 10

ndash 17 ndash

JCAP01(2016)028

-800 -600 -400 -200 00

GrowthRateκ

5101520

-150 -100 -50 0 50 100 1500

GrowthRateκ

1010 2020

-3000 -2500 -2000 -1500 -1000 -500 00

GrowthRateκ

255075100

-400 -200 0 200 4000

GrowthRateκ

2525 5050 7575 100100

ndash 18 ndash

JCAP01(2016)028

I2 0 I4

0 minus2I2 0I0 0 I2

= 0 (321)

ndash 19 ndash

JCAP01(2016)028

GrowthRateκ

k = 25 nv = 2

GrowthRateκ

k = 25 nv = 5

GrowthRateκ

k = 25 nv = 10

-1200-1000 -800 -600 -400 -200 0 2000

GrowthRateκ

k = 25 nv = 20

ndash 20 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

ndash 21 ndash

JCAP01(2016)028

int +infin

int +π

ndash 22 ndash

JCAP01(2016)028

λndash

k = 01041

k = 0 102 103 104

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

= 0 (43)

Ia = micro

int +infin

minusinfindω hω

int +π

minusπdϕ

fa(ϕ)

ndash 23 ndash

JCAP01(2016)028

1 0 00 minus1

2 00 0 minus1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (45)

Ia =micro

int +infin

minusinfindω hωFa

(ω minus Ω

)where Fa(w) =

int +π

minusπdϕ

fa(ϕ)

cosϕ+ w (46)

Ia =micro

(ω0 minus Ω

)minus αFa

(minusω0 minus Ω

)] (47)

radicw + 1 (48)

s(w) Fc = 1minus w

s(w) Fcc = minusw +

ndash 24 ndash

JCAP01(2016)028

-3000 -2000 -1000 0 1000 2000

micro (410)

radicω0

radicω0 minus Ω

ndash 25 ndash

JCAP01(2016)028

intdv = (1π)

int +πminusπ dϕ

= micro

int +infin

int +π

minusπ

dϕprime

0dvprime vprime

QΩkωvϕ =A0 +A2v

2 + k v cϕ + ω minus Ω (412)

3 I15 Ic

1 I13 Ic

2 0minus2Ic

3 00 0 0 minus2Iss

= 0 (413)

integrals are now

Ian = micro

int +infin

minusinfindω hω

int +π

minusπ

vn fa(ϕ)12 λv

2 + k v cϕ + ω minus Ω (414)

3 = I13 minus Icc

= 0 (415)

In = micro

int +infin

12 λv

2 + ω minus Ω (416)

ndash 26 ndash

JCAP01(2016)028

2(ω minus Ω)

) (417a)

2(ω minus Ω)

)](417b)

(2(ω minus Ω)

2(ω minus Ω)

)] (417c)

127 lt microω0 lt 4328 (419a)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v cϕ + w(420)

K11 = w +

radic1minus w

(421a)

radicminuswradic

1minus w2 (1 + 2w2)

3radicw

(421b)

radicminuswradic

1minus w2 (3 + 4w2 + 8w4)

15radicw

(421c)

2minus w2 minus

radicminusw2

ndash 27 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

4minus 2w4 +

radicminusw2radic

1minus w2 (1 + 2w2)

3 (421e)

Kcc3 = minusw

2minus w2 minus

radicminusw2

radic1minus w2

) (421f)

Kss3 =

w(3minus 2w2)

3radicw

(421g)

3 = K13

ndash 28 ndash

JCAP01(2016)028

-1000 -500 0 500

micro = ai

radick)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

-800 -600 -400 -200 00

GrowthRateκ

5101520

-150 -100 -50 0 50 100 1500

GrowthRateκ

1010 2020

-3000 -2500 -2000 -1500 -1000 -500 00

GrowthRateκ

255075100

-400 -200 0 200 4000

GrowthRateκ

2525 5050 7575 100100

ndash 18 ndash

JCAP01(2016)028

I2 0 I4

0 minus2I2 0I0 0 I2

= 0 (321)

ndash 19 ndash

JCAP01(2016)028

GrowthRateκ

k = 25 nv = 2

GrowthRateκ

k = 25 nv = 5

GrowthRateκ

k = 25 nv = 10

-1200-1000 -800 -600 -400 -200 0 2000

GrowthRateκ

k = 25 nv = 20

ndash 20 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

ndash 21 ndash

JCAP01(2016)028

int +infin

int +π

ndash 22 ndash

JCAP01(2016)028

λndash

k = 01041

k = 0 102 103 104

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

= 0 (43)

Ia = micro

int +infin

minusinfindω hω

int +π

minusπdϕ

fa(ϕ)

ndash 23 ndash

JCAP01(2016)028

1 0 00 minus1

2 00 0 minus1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (45)

Ia =micro

int +infin

minusinfindω hωFa

(ω minus Ω

)where Fa(w) =

int +π

minusπdϕ

fa(ϕ)

cosϕ+ w (46)

Ia =micro

(ω0 minus Ω

)minus αFa

(minusω0 minus Ω

)] (47)

radicw + 1 (48)

s(w) Fc = 1minus w

s(w) Fcc = minusw +

ndash 24 ndash

JCAP01(2016)028

-3000 -2000 -1000 0 1000 2000

micro (410)

radicω0

radicω0 minus Ω

ndash 25 ndash

JCAP01(2016)028

intdv = (1π)

int +πminusπ dϕ

= micro

int +infin

int +π

minusπ

dϕprime

0dvprime vprime

QΩkωvϕ =A0 +A2v

2 + k v cϕ + ω minus Ω (412)

3 I15 Ic

1 I13 Ic

2 0minus2Ic

3 00 0 0 minus2Iss

= 0 (413)

integrals are now

Ian = micro

int +infin

minusinfindω hω

int +π

minusπ

vn fa(ϕ)12 λv

2 + k v cϕ + ω minus Ω (414)

3 = I13 minus Icc

= 0 (415)

In = micro

int +infin

12 λv

2 + ω minus Ω (416)

ndash 26 ndash

JCAP01(2016)028

2(ω minus Ω)

) (417a)

2(ω minus Ω)

)](417b)

(2(ω minus Ω)

2(ω minus Ω)

)] (417c)

127 lt microω0 lt 4328 (419a)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v cϕ + w(420)

K11 = w +

radic1minus w

(421a)

radicminuswradic

1minus w2 (1 + 2w2)

3radicw

(421b)

radicminuswradic

1minus w2 (3 + 4w2 + 8w4)

15radicw

(421c)

2minus w2 minus

radicminusw2

ndash 27 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

4minus 2w4 +

radicminusw2radic

1minus w2 (1 + 2w2)

3 (421e)

Kcc3 = minusw

2minus w2 minus

radicminusw2

radic1minus w2

) (421f)

Kss3 =

w(3minus 2w2)

3radicw

(421g)

3 = K13

ndash 28 ndash

JCAP01(2016)028

-1000 -500 0 500

micro = ai

radick)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

I2 0 I4

0 minus2I2 0I0 0 I2

= 0 (321)

ndash 19 ndash

JCAP01(2016)028

GrowthRateκ

k = 25 nv = 2

GrowthRateκ

k = 25 nv = 5

GrowthRateκ

k = 25 nv = 10

-1200-1000 -800 -600 -400 -200 0 2000

GrowthRateκ

k = 25 nv = 20

ndash 20 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

ndash 21 ndash

JCAP01(2016)028

int +infin

int +π

ndash 22 ndash

JCAP01(2016)028

λndash

k = 01041

k = 0 102 103 104

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

= 0 (43)

Ia = micro

int +infin

minusinfindω hω

int +π

minusπdϕ

fa(ϕ)

ndash 23 ndash

JCAP01(2016)028

1 0 00 minus1

2 00 0 minus1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (45)

Ia =micro

int +infin

minusinfindω hωFa

(ω minus Ω

)where Fa(w) =

int +π

minusπdϕ

fa(ϕ)

cosϕ+ w (46)

Ia =micro

(ω0 minus Ω

)minus αFa

(minusω0 minus Ω

)] (47)

radicw + 1 (48)

s(w) Fc = 1minus w

s(w) Fcc = minusw +

ndash 24 ndash

JCAP01(2016)028

-3000 -2000 -1000 0 1000 2000

micro (410)

radicω0

radicω0 minus Ω

ndash 25 ndash

JCAP01(2016)028

intdv = (1π)

int +πminusπ dϕ

= micro

int +infin

int +π

minusπ

dϕprime

0dvprime vprime

QΩkωvϕ =A0 +A2v

2 + k v cϕ + ω minus Ω (412)

3 I15 Ic

1 I13 Ic

2 0minus2Ic

3 00 0 0 minus2Iss

= 0 (413)

integrals are now

Ian = micro

int +infin

minusinfindω hω

int +π

minusπ

vn fa(ϕ)12 λv

2 + k v cϕ + ω minus Ω (414)

3 = I13 minus Icc

= 0 (415)

In = micro

int +infin

12 λv

2 + ω minus Ω (416)

ndash 26 ndash

JCAP01(2016)028

2(ω minus Ω)

) (417a)

2(ω minus Ω)

)](417b)

(2(ω minus Ω)

2(ω minus Ω)

)] (417c)

127 lt microω0 lt 4328 (419a)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v cϕ + w(420)

K11 = w +

radic1minus w

(421a)

radicminuswradic

1minus w2 (1 + 2w2)

3radicw

(421b)

radicminuswradic

1minus w2 (3 + 4w2 + 8w4)

15radicw

(421c)

2minus w2 minus

radicminusw2

ndash 27 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

4minus 2w4 +

radicminusw2radic

1minus w2 (1 + 2w2)

3 (421e)

Kcc3 = minusw

2minus w2 minus

radicminusw2

radic1minus w2

) (421f)

Kss3 =

w(3minus 2w2)

3radicw

(421g)

3 = K13

ndash 28 ndash

JCAP01(2016)028

-1000 -500 0 500

micro = ai

radick)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

GrowthRateκ

k = 25 nv = 2

GrowthRateκ

k = 25 nv = 5

GrowthRateκ

k = 25 nv = 10

-1200-1000 -800 -600 -400 -200 0 2000

GrowthRateκ

k = 25 nv = 20

ndash 20 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

ndash 21 ndash

JCAP01(2016)028

int +infin

int +π

ndash 22 ndash

JCAP01(2016)028

λndash

k = 01041

k = 0 102 103 104

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

= 0 (43)

Ia = micro

int +infin

minusinfindω hω

int +π

minusπdϕ

fa(ϕ)

ndash 23 ndash

JCAP01(2016)028

1 0 00 minus1

2 00 0 minus1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (45)

Ia =micro

int +infin

minusinfindω hωFa

(ω minus Ω

)where Fa(w) =

int +π

minusπdϕ

fa(ϕ)

cosϕ+ w (46)

Ia =micro

(ω0 minus Ω

)minus αFa

(minusω0 minus Ω

)] (47)

radicw + 1 (48)

s(w) Fc = 1minus w

s(w) Fcc = minusw +

ndash 24 ndash

JCAP01(2016)028

-3000 -2000 -1000 0 1000 2000

micro (410)

radicω0

radicω0 minus Ω

ndash 25 ndash

JCAP01(2016)028

intdv = (1π)

int +πminusπ dϕ

= micro

int +infin

int +π

minusπ

dϕprime

0dvprime vprime

QΩkωvϕ =A0 +A2v

2 + k v cϕ + ω minus Ω (412)

3 I15 Ic

1 I13 Ic

2 0minus2Ic

3 00 0 0 minus2Iss

= 0 (413)

integrals are now

Ian = micro

int +infin

minusinfindω hω

int +π

minusπ

vn fa(ϕ)12 λv

2 + k v cϕ + ω minus Ω (414)

3 = I13 minus Icc

= 0 (415)

In = micro

int +infin

12 λv

2 + ω minus Ω (416)

ndash 26 ndash

JCAP01(2016)028

2(ω minus Ω)

) (417a)

2(ω minus Ω)

)](417b)

(2(ω minus Ω)

2(ω minus Ω)

)] (417c)

127 lt microω0 lt 4328 (419a)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v cϕ + w(420)

K11 = w +

radic1minus w

(421a)

radicminuswradic

1minus w2 (1 + 2w2)

3radicw

(421b)

radicminuswradic

1minus w2 (3 + 4w2 + 8w4)

15radicw

(421c)

2minus w2 minus

radicminusw2

ndash 27 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

4minus 2w4 +

radicminusw2radic

1minus w2 (1 + 2w2)

3 (421e)

Kcc3 = minusw

2minus w2 minus

radicminusw2

radic1minus w2

) (421f)

Kss3 =

w(3minus 2w2)

3radicw

(421g)

3 = K13

ndash 28 ndash

JCAP01(2016)028

-1000 -500 0 500

micro = ai

radick)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

ndash 21 ndash

JCAP01(2016)028

int +infin

int +π

ndash 22 ndash

JCAP01(2016)028

λndash

k = 01041

k = 0 102 103 104

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

= 0 (43)

Ia = micro

int +infin

minusinfindω hω

int +π

minusπdϕ

fa(ϕ)

ndash 23 ndash

JCAP01(2016)028

1 0 00 minus1

2 00 0 minus1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (45)

Ia =micro

int +infin

minusinfindω hωFa

(ω minus Ω

)where Fa(w) =

int +π

minusπdϕ

fa(ϕ)

cosϕ+ w (46)

Ia =micro

(ω0 minus Ω

)minus αFa

(minusω0 minus Ω

)] (47)

radicw + 1 (48)

s(w) Fc = 1minus w

s(w) Fcc = minusw +

ndash 24 ndash

JCAP01(2016)028

-3000 -2000 -1000 0 1000 2000

micro (410)

radicω0

radicω0 minus Ω

ndash 25 ndash

JCAP01(2016)028

intdv = (1π)

int +πminusπ dϕ

= micro

int +infin

int +π

minusπ

dϕprime

0dvprime vprime

QΩkωvϕ =A0 +A2v

2 + k v cϕ + ω minus Ω (412)

3 I15 Ic

1 I13 Ic

2 0minus2Ic

3 00 0 0 minus2Iss

= 0 (413)

integrals are now

Ian = micro

int +infin

minusinfindω hω

int +π

minusπ

vn fa(ϕ)12 λv

2 + k v cϕ + ω minus Ω (414)

3 = I13 minus Icc

= 0 (415)

In = micro

int +infin

12 λv

2 + ω minus Ω (416)

ndash 26 ndash

JCAP01(2016)028

2(ω minus Ω)

) (417a)

2(ω minus Ω)

)](417b)

(2(ω minus Ω)

2(ω minus Ω)

)] (417c)

127 lt microω0 lt 4328 (419a)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v cϕ + w(420)

K11 = w +

radic1minus w

(421a)

radicminuswradic

1minus w2 (1 + 2w2)

3radicw

(421b)

radicminuswradic

1minus w2 (3 + 4w2 + 8w4)

15radicw

(421c)

2minus w2 minus

radicminusw2

ndash 27 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

4minus 2w4 +

radicminusw2radic

1minus w2 (1 + 2w2)

3 (421e)

Kcc3 = minusw

2minus w2 minus

radicminusw2

radic1minus w2

) (421f)

Kss3 =

w(3minus 2w2)

3radicw

(421g)

3 = K13

ndash 28 ndash

JCAP01(2016)028

-1000 -500 0 500

micro = ai

radick)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

int +infin

int +π

ndash 22 ndash

JCAP01(2016)028

λndash

k = 01041

k = 0 102 103 104

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

= 0 (43)

Ia = micro

int +infin

minusinfindω hω

int +π

minusπdϕ

fa(ϕ)

ndash 23 ndash

JCAP01(2016)028

1 0 00 minus1

2 00 0 minus1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (45)

Ia =micro

int +infin

minusinfindω hωFa

(ω minus Ω

)where Fa(w) =

int +π

minusπdϕ

fa(ϕ)

cosϕ+ w (46)

Ia =micro

(ω0 minus Ω

)minus αFa

(minusω0 minus Ω

)] (47)

radicw + 1 (48)

s(w) Fc = 1minus w

s(w) Fcc = minusw +

ndash 24 ndash

JCAP01(2016)028

-3000 -2000 -1000 0 1000 2000

micro (410)

radicω0

radicω0 minus Ω

ndash 25 ndash

JCAP01(2016)028

intdv = (1π)

int +πminusπ dϕ

= micro

int +infin

int +π

minusπ

dϕprime

0dvprime vprime

QΩkωvϕ =A0 +A2v

2 + k v cϕ + ω minus Ω (412)

3 I15 Ic

1 I13 Ic

2 0minus2Ic

3 00 0 0 minus2Iss

= 0 (413)

integrals are now

Ian = micro

int +infin

minusinfindω hω

int +π

minusπ

vn fa(ϕ)12 λv

2 + k v cϕ + ω minus Ω (414)

3 = I13 minus Icc

= 0 (415)

In = micro

int +infin

12 λv

2 + ω minus Ω (416)

ndash 26 ndash

JCAP01(2016)028

2(ω minus Ω)

) (417a)

2(ω minus Ω)

)](417b)

(2(ω minus Ω)

2(ω minus Ω)

)] (417c)

127 lt microω0 lt 4328 (419a)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v cϕ + w(420)

K11 = w +

radic1minus w

(421a)

radicminuswradic

1minus w2 (1 + 2w2)

3radicw

(421b)

radicminuswradic

1minus w2 (3 + 4w2 + 8w4)

15radicw

(421c)

2minus w2 minus

radicminusw2

ndash 27 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

4minus 2w4 +

radicminusw2radic

1minus w2 (1 + 2w2)

3 (421e)

Kcc3 = minusw

2minus w2 minus

radicminusw2

radic1minus w2

) (421f)

Kss3 =

w(3minus 2w2)

3radicw

(421g)

3 = K13

ndash 28 ndash

JCAP01(2016)028

-1000 -500 0 500

micro = ai

radick)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

λndash

k = 01041

k = 0 102 103 104

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

= 0 (43)

Ia = micro

int +infin

minusinfindω hω

int +π

minusπdϕ

fa(ϕ)

ndash 23 ndash

JCAP01(2016)028

1 0 00 minus1

2 00 0 minus1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (45)

Ia =micro

int +infin

minusinfindω hωFa

(ω minus Ω

)where Fa(w) =

int +π

minusπdϕ

fa(ϕ)

cosϕ+ w (46)

Ia =micro

(ω0 minus Ω

)minus αFa

(minusω0 minus Ω

)] (47)

radicw + 1 (48)

s(w) Fc = 1minus w

s(w) Fcc = minusw +

ndash 24 ndash

JCAP01(2016)028

-3000 -2000 -1000 0 1000 2000

micro (410)

radicω0

radicω0 minus Ω

ndash 25 ndash

JCAP01(2016)028

intdv = (1π)

int +πminusπ dϕ

= micro

int +infin

int +π

minusπ

dϕprime

0dvprime vprime

QΩkωvϕ =A0 +A2v

2 + k v cϕ + ω minus Ω (412)

3 I15 Ic

1 I13 Ic

2 0minus2Ic

3 00 0 0 minus2Iss

= 0 (413)

integrals are now

Ian = micro

int +infin

minusinfindω hω

int +π

minusπ

vn fa(ϕ)12 λv

2 + k v cϕ + ω minus Ω (414)

3 = I13 minus Icc

= 0 (415)

In = micro

int +infin

12 λv

2 + ω minus Ω (416)

ndash 26 ndash

JCAP01(2016)028

2(ω minus Ω)

) (417a)

2(ω minus Ω)

)](417b)

(2(ω minus Ω)

2(ω minus Ω)

)] (417c)

127 lt microω0 lt 4328 (419a)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v cϕ + w(420)

K11 = w +

radic1minus w

(421a)

radicminuswradic

1minus w2 (1 + 2w2)

3radicw

(421b)

radicminuswradic

1minus w2 (3 + 4w2 + 8w4)

15radicw

(421c)

2minus w2 minus

radicminusw2

ndash 27 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

4minus 2w4 +

radicminusw2radic

1minus w2 (1 + 2w2)

3 (421e)

Kcc3 = minusw

2minus w2 minus

radicminusw2

radic1minus w2

) (421f)

Kss3 =

w(3minus 2w2)

3radicw

(421g)

3 = K13

ndash 28 ndash

JCAP01(2016)028

-1000 -500 0 500

micro = ai

radick)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

1 0 00 minus1

2 00 0 minus1

[(1 + α)ω0 + (1minus α) Ω]

= 0 (45)

Ia =micro

int +infin

minusinfindω hωFa

(ω minus Ω

)where Fa(w) =

int +π

minusπdϕ

fa(ϕ)

cosϕ+ w (46)

Ia =micro

(ω0 minus Ω

)minus αFa

(minusω0 minus Ω

)] (47)

radicw + 1 (48)

s(w) Fc = 1minus w

s(w) Fcc = minusw +

ndash 24 ndash

JCAP01(2016)028

-3000 -2000 -1000 0 1000 2000

micro (410)

radicω0

radicω0 minus Ω

ndash 25 ndash

JCAP01(2016)028

intdv = (1π)

int +πminusπ dϕ

= micro

int +infin

int +π

minusπ

dϕprime

0dvprime vprime

QΩkωvϕ =A0 +A2v

2 + k v cϕ + ω minus Ω (412)

3 I15 Ic

1 I13 Ic

2 0minus2Ic

3 00 0 0 minus2Iss

= 0 (413)

integrals are now

Ian = micro

int +infin

minusinfindω hω

int +π

minusπ

vn fa(ϕ)12 λv

2 + k v cϕ + ω minus Ω (414)

3 = I13 minus Icc

= 0 (415)

In = micro

int +infin

12 λv

2 + ω minus Ω (416)

ndash 26 ndash

JCAP01(2016)028

2(ω minus Ω)

) (417a)

2(ω minus Ω)

)](417b)

(2(ω minus Ω)

2(ω minus Ω)

)] (417c)

127 lt microω0 lt 4328 (419a)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v cϕ + w(420)

K11 = w +

radic1minus w

(421a)

radicminuswradic

1minus w2 (1 + 2w2)

3radicw

(421b)

radicminuswradic

1minus w2 (3 + 4w2 + 8w4)

15radicw

(421c)

2minus w2 minus

radicminusw2

ndash 27 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

4minus 2w4 +

radicminusw2radic

1minus w2 (1 + 2w2)

3 (421e)

Kcc3 = minusw

2minus w2 minus

radicminusw2

radic1minus w2

) (421f)

Kss3 =

w(3minus 2w2)

3radicw

(421g)

3 = K13

ndash 28 ndash

JCAP01(2016)028

-1000 -500 0 500

micro = ai

radick)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

-3000 -2000 -1000 0 1000 2000

micro (410)

radicω0

radicω0 minus Ω

ndash 25 ndash

JCAP01(2016)028

intdv = (1π)

int +πminusπ dϕ

= micro

int +infin

int +π

minusπ

dϕprime

0dvprime vprime

QΩkωvϕ =A0 +A2v

2 + k v cϕ + ω minus Ω (412)

3 I15 Ic

1 I13 Ic

2 0minus2Ic

3 00 0 0 minus2Iss

= 0 (413)

integrals are now

Ian = micro

int +infin

minusinfindω hω

int +π

minusπ

vn fa(ϕ)12 λv

2 + k v cϕ + ω minus Ω (414)

3 = I13 minus Icc

= 0 (415)

In = micro

int +infin

12 λv

2 + ω minus Ω (416)

ndash 26 ndash

JCAP01(2016)028

2(ω minus Ω)

) (417a)

2(ω minus Ω)

)](417b)

(2(ω minus Ω)

2(ω minus Ω)

)] (417c)

127 lt microω0 lt 4328 (419a)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v cϕ + w(420)

K11 = w +

radic1minus w

(421a)

radicminuswradic

1minus w2 (1 + 2w2)

3radicw

(421b)

radicminuswradic

1minus w2 (3 + 4w2 + 8w4)

15radicw

(421c)

2minus w2 minus

radicminusw2

ndash 27 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

4minus 2w4 +

radicminusw2radic

1minus w2 (1 + 2w2)

3 (421e)

Kcc3 = minusw

2minus w2 minus

radicminusw2

radic1minus w2

) (421f)

Kss3 =

w(3minus 2w2)

3radicw

(421g)

3 = K13

ndash 28 ndash

JCAP01(2016)028

-1000 -500 0 500

micro = ai

radick)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

intdv = (1π)

int +πminusπ dϕ

= micro

int +infin

int +π

minusπ

dϕprime

0dvprime vprime

QΩkωvϕ =A0 +A2v

2 + k v cϕ + ω minus Ω (412)

3 I15 Ic

1 I13 Ic

2 0minus2Ic

3 00 0 0 minus2Iss

= 0 (413)

integrals are now

Ian = micro

int +infin

minusinfindω hω

int +π

minusπ

vn fa(ϕ)12 λv

2 + k v cϕ + ω minus Ω (414)

3 = I13 minus Icc

= 0 (415)

In = micro

int +infin

12 λv

2 + ω minus Ω (416)

ndash 26 ndash

JCAP01(2016)028

2(ω minus Ω)

) (417a)

2(ω minus Ω)

)](417b)

(2(ω minus Ω)

2(ω minus Ω)

)] (417c)

127 lt microω0 lt 4328 (419a)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v cϕ + w(420)

K11 = w +

radic1minus w

(421a)

radicminuswradic

1minus w2 (1 + 2w2)

3radicw

(421b)

radicminuswradic

1minus w2 (3 + 4w2 + 8w4)

15radicw

(421c)

2minus w2 minus

radicminusw2

ndash 27 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

4minus 2w4 +

radicminusw2radic

1minus w2 (1 + 2w2)

3 (421e)

Kcc3 = minusw

2minus w2 minus

radicminusw2

radic1minus w2

) (421f)

Kss3 =

w(3minus 2w2)

3radicw

(421g)

3 = K13

ndash 28 ndash

JCAP01(2016)028

-1000 -500 0 500

micro = ai

radick)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

2(ω minus Ω)

) (417a)

2(ω minus Ω)

)](417b)

(2(ω minus Ω)

2(ω minus Ω)

)] (417c)

127 lt microω0 lt 4328 (419a)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v cϕ + w(420)

K11 = w +

radic1minus w

(421a)

radicminuswradic

1minus w2 (1 + 2w2)

3radicw

(421b)

radicminuswradic

1minus w2 (3 + 4w2 + 8w4)

15radicw

(421c)

2minus w2 minus

radicminusw2

ndash 27 ndash

JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

4minus 2w4 +

radicminusw2radic

1minus w2 (1 + 2w2)

3 (421e)

Kcc3 = minusw

2minus w2 minus

radicminusw2

radic1minus w2

) (421f)

Kss3 =

w(3minus 2w2)

3radicw

(421g)

3 = K13

ndash 28 ndash

JCAP01(2016)028

-1000 -500 0 500

micro = ai

radick)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

BimodalInstability

MZA Instability

GrowthRateκ

-1200 -1000 -800 -600 -400 -200 0 200 400

λndash

MAA Instability

GrowthRateκ

4minus 2w4 +

radicminusw2radic

1minus w2 (1 + 2w2)

3 (421e)

Kcc3 = minusw

2minus w2 minus

radicminusw2

radic1minus w2

) (421f)

Kss3 =

w(3minus 2w2)

3radicw

(421g)

3 = K13

ndash 28 ndash

JCAP01(2016)028

-1000 -500 0 500

micro = ai

radick)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

-1000 -500 0 500

micro = ai

radick)

Ian =micro

int +infin

minusinfindω hωK

an where Ka

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (423)

ndash 29 ndash

JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

tyλndash

- - - - - - -

ndash 30 ndash

JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

5 Conclusions

ndash 31 ndash

JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

Acknowledgments

fn(w) =

int +1

minus1dv

v + w (A1)

L(w) = log

(w + 1

w minus 1

) (A2a)

] (A2b)

f0(w) = L(w) (A3a)

f1(w) = A(w) (A3b)

f3(w) =2

3+ w2A(w) (A3d)

f4(w) = minus2

(w + 3w3

)+ w4L(w) (A3e)

the form

gn(p w) =

int +1

minus1dv

v2 + p v + w (A4)

K(p w) =1

(w + 1 + p

w + 1minus p

) (A5a)

B(p w) =1radic

4w minus p2

[arctan

)+ arctan

)] (A5b)

ndash 32 ndash

JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

g0(p w) = 2B(p w) (A6a)

)B(p w) +

(p2 minus w

)K(p w) (A6d)

g4(p w) =2

(1 + 3p2 minus 3w

)B(p w) minus

(p3 minus 2pw

)K(p w) (A6e)

int +π

minusπ

vn fa(ϕ)

v22 + q v cϕ + w (A7)

Aqw = arctan

)+ arctan

) (A8a)

K11 = Aqw Sw (A9a)

[Bqw + 2

(q2 minus w

]Sw (A9b)

(6q2 minus 6w + 1

)Bqw + 4

] Sw2 (A9c)

1minus(Bqw + 2q2Aqw

2q (A9d)

(6q2 minus 2w + 1

)Bqw minus 4q2

(3q2 minus 4w

4q (A9e)

Kcc3 =

(6q2 + 2w + 1

)Bqw minus 4q2

(3q2 minus 2w

8q2 (A9f)

(2micro

ω0 minus Ω

)minus αF

(2micro

minusω0 minus Ω

)= 1minus α (B1)

ndash 33 ndash

JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

micro = mω01 + α

1 + α

1minus α (B2)

leading to

)minus αF

)= 1minus α (B3)

w = minusmplusmn

(1minus α1 + α

(1minusradicα)2

1 + αlt m lt

(1 +radicα)2

1 + α (B5)

K1(αm) =

(1minus α1 + α

Kmax1 =

2radicα

1 + α (B7)

ndash 34 ndash

JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

00 05 10 15 2000

mK1(αm

F(x) = x

00 05 10 1500

K12(αm

F(x) = x

0 1 2 300

Klog(αm

F(x) = log(x)

minus α(

= 1minus α (B8)

K12(1m) =8m (3minus 2m) + (2m)23

[8m (9minus 4m) + 3

]234radic

3 (2m)13[8m (9minus 4m) + 3

]13 (B9)

value of 116

radic32(69 + 11

radic33) = 0880086 at m = 3

32(3 +radic

33) = 0819803 In figure 12 we

ndash 35 ndash

JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

)minus α log

)= 1minus α (B10)

log = 0724611

radic3m(3mminus 4e)

3e (B11)

micro =micro

1minus α(6π)12 (2ω0)14 λ34 (C1)

radicω0

minusω0 minus Ω=

1minus αmicro2

ndash 36 ndash

JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

-3 -2 -1 0 1 200

μ prop μ λ34

GrowthRate

102 104 106 infin

10-1 10010-5

μ prop μ λ34

GrowthRate

λ = infin

micro =minusλ

radic1minus α2

2 (1minus α)2

radicω0λ (C4)

1minus α2

ndash 37 ndash

JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

00 05 10 1500

GrowthRate

102103λ = infin

3ω0λ2 (C6)

1minus α= a (C7)

a = log

(minus 2λ

)minus 1

microor a = log

(minus 2λ

3 + micro2

radicea(8minus ea)

2 (C9)

ndash 38 ndash

JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

(minus2λ

)= log

(minusλ

2e2ω0

) (C11)

micro = minus2λ6

3A+radic

A (C12a)

micro = +2λ1

A (C12b)

micro = +2λ3A+

radic3999950) = minus990348

3999950) (C13)

ndash 39 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

minus 3 i

[radic2a2(32 + a)

radicx+ 1

)] 1radick

+O(1k) (D2)

ndash 40 ndash

JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

a1 = minus371825 (D3)

a2 = minus342115 (D4)

a3 = +205821 (D5)

mmax =162radic

3 [120minus a (20 + 3a)]

11 [1080minus a (120 + 53a)] (D6)

mmax1 = 123675 (D7)

mmax2 = 449396 (D8)

mmax3 = 277208 (D9)

6m = immax

radicx+ 1

) (D10)

3minus 10m2

3m2max

(k +mmaxi

radick) (D12)

6(minus9 + b+ 3x)

+ i2radic

k32+O(1k2) (D14)

kminus i 4radic

k32+O(1k2) (D15)

ndash 41 ndash

JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

] (D16)

radick Therefore

micro =micro

1minus αλ (E1)

micro2 (E2)

1minus α (E3)

a = log

(minus λ

)minus micro+ 1

micro (E4)

ndash 42 ndash

JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

λndash

-106 -105 -104 -103 -102 -10 -1

1 10 102 103 104 105 106

0 2 4 6 800

GrowthRate

κ λ = 102

ndash 43 ndash

JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

References

ndash 44 ndash

JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

ndash 45 ndash

JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

ndash 46 ndash

JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

ndash 47 ndash

JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





JCAP01(2016)028

ndash 48 ndash

Introduction

Equations of motion




Two-flavor system

Mass ordering

Linearization







Eigenvalue equation




Eigenvalue equation








Eigenvalue equation




Eigenvalue equation




Conclusions

Analytic integrals





Documents

Self-induced avor conversion of supernova neutrinos on small … · 2016. 1. 18. · JCAP01(2016)028 ou rnal of C osmology and A strop ar ticle P hysics J An IOP and SISSA journal